Properties

Label 1-1700-1700.1031-r0-0-0
Degree 11
Conductor 17001700
Sign 0.623+0.781i-0.623 + 0.781i
Analytic cond. 7.894767.89476
Root an. cond. 7.894767.89476
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0784 − 0.996i)3-s + (−0.382 + 0.923i)7-s + (−0.987 + 0.156i)9-s + (0.972 − 0.233i)11-s + (−0.587 + 0.809i)13-s + (0.453 − 0.891i)19-s + (0.951 + 0.309i)21-s + (−0.972 + 0.233i)23-s + (0.233 + 0.972i)27-s + (−0.996 + 0.0784i)29-s + (−0.649 − 0.760i)31-s + (−0.309 − 0.951i)33-s + (0.972 + 0.233i)37-s + (0.852 + 0.522i)39-s + (−0.852 + 0.522i)41-s + ⋯
L(s)  = 1  + (−0.0784 − 0.996i)3-s + (−0.382 + 0.923i)7-s + (−0.987 + 0.156i)9-s + (0.972 − 0.233i)11-s + (−0.587 + 0.809i)13-s + (0.453 − 0.891i)19-s + (0.951 + 0.309i)21-s + (−0.972 + 0.233i)23-s + (0.233 + 0.972i)27-s + (−0.996 + 0.0784i)29-s + (−0.649 − 0.760i)31-s + (−0.309 − 0.951i)33-s + (0.972 + 0.233i)37-s + (0.852 + 0.522i)39-s + (−0.852 + 0.522i)41-s + ⋯

Functional equation

Λ(s)=(1700s/2ΓR(s)L(s)=((0.623+0.781i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.623 + 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1700s/2ΓR(s)L(s)=((0.623+0.781i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.623 + 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 17001700    =    2252172^{2} \cdot 5^{2} \cdot 17
Sign: 0.623+0.781i-0.623 + 0.781i
Analytic conductor: 7.894767.89476
Root analytic conductor: 7.894767.89476
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1700(1031,)\chi_{1700} (1031, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 1700, (0: ), 0.623+0.781i)(1,\ 1700,\ (0:\ ),\ -0.623 + 0.781i)

Particular Values

L(12)L(\frac{1}{2}) \approx 0.1048551655+0.2178954245i0.1048551655 + 0.2178954245i
L(12)L(\frac12) \approx 0.1048551655+0.2178954245i0.1048551655 + 0.2178954245i
L(1)L(1) \approx 0.77503444500.1177339012i0.7750344450 - 0.1177339012i
L(1)L(1) \approx 0.77503444500.1177339012i0.7750344450 - 0.1177339012i

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
5 1 1
17 1 1
good3 1+(0.07840.996i)T 1 + (-0.0784 - 0.996i)T
7 1+(0.382+0.923i)T 1 + (-0.382 + 0.923i)T
11 1+(0.9720.233i)T 1 + (0.972 - 0.233i)T
13 1+(0.587+0.809i)T 1 + (-0.587 + 0.809i)T
19 1+(0.4530.891i)T 1 + (0.453 - 0.891i)T
23 1+(0.972+0.233i)T 1 + (-0.972 + 0.233i)T
29 1+(0.996+0.0784i)T 1 + (-0.996 + 0.0784i)T
31 1+(0.6490.760i)T 1 + (-0.649 - 0.760i)T
37 1+(0.972+0.233i)T 1 + (0.972 + 0.233i)T
41 1+(0.852+0.522i)T 1 + (-0.852 + 0.522i)T
43 1+(0.707+0.707i)T 1 + (-0.707 + 0.707i)T
47 1+(0.951+0.309i)T 1 + (0.951 + 0.309i)T
53 1+(0.8910.453i)T 1 + (0.891 - 0.453i)T
59 1+(0.987+0.156i)T 1 + (-0.987 + 0.156i)T
61 1+(0.2330.972i)T 1 + (-0.233 - 0.972i)T
67 1+(0.309+0.951i)T 1 + (0.309 + 0.951i)T
71 1+(0.07840.996i)T 1 + (-0.0784 - 0.996i)T
73 1+(0.8520.522i)T 1 + (-0.852 - 0.522i)T
79 1+(0.649+0.760i)T 1 + (-0.649 + 0.760i)T
83 1+(0.453+0.891i)T 1 + (-0.453 + 0.891i)T
89 1+(0.5870.809i)T 1 + (-0.587 - 0.809i)T
97 1+(0.996+0.0784i)T 1 + (-0.996 + 0.0784i)T
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   L(s)=p (1αpps)1L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−20.21732824449432104506578990221, −19.69226451622295386370408234891, −18.56091951991490065723050405925, −17.62201961895278327808480242200, −16.87694125997421713885329751284, −16.54385856126275643629495589270, −15.64226060319783044365655236084, −14.8203605443077300537760982967, −14.26911369881854231690868857044, −13.4990687168156024640627964358, −12.39560873940017563455580637453, −11.79415402995523956013295803417, −10.765834296750277134208072888452, −10.17917939495959482581830766573, −9.627963807385288601582066788867, −8.78638620257080567571544312909, −7.781743726307408088509851454309, −7.00656555530503745410469379654, −5.96042624306117226683205627290, −5.27796275066275678313981842426, −4.11662604504260336042768384704, −3.795886982602081835979313766948, −2.81615508975504654480150001252, −1.47864122577361435312410000328, −0.08619618612333397573105566731, 1.38493900773902843163118548083, 2.1923387761897094386256235958, 3.00867115872285093512379527068, 4.102181295306073192146290363450, 5.27230655932801701422337790773, 6.04586929056525391598076826054, 6.69024825235687523602498859982, 7.45366324052547180653694794512, 8.37204036102776438717951832204, 9.22474291248931553967797859615, 9.65537541496923803921190875981, 11.211473127568846691856417638357, 11.66471577479899614877881213730, 12.25701096132281428938997575130, 13.1014974963335004902351572689, 13.782772951711434407401880981337, 14.60071136718704529915361368352, 15.229128267219606539119849804440, 16.40623680766313733954499425807, 16.85982471328313734766846315810, 17.783669008903321471098641497349, 18.49017574874518273328310699267, 19.00585983830147298372740930063, 19.80868116928908224781212501745, 20.21970572625811836341750565657

Graph of the ZZ-function along the critical line