↑

Properties

Label 1-1700-1700.1031-r0-0-0
Degree $1$
Conductor $1700$
Sign $-0.623 + 0.781i$
Analytic cond. $7.89476$
Root an. cond. $7.89476$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

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

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

Invariants

Degree: \(1\)
Conductor: \(1700\)    =    \(2^{2} \cdot 5^{2} \cdot 17\)
Sign: $-0.623 + 0.781i$
Analytic conductor: \(7.89476\)
Root analytic conductor: \(7.89476\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1700} (1031, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1700,\ (0:\ ),\ -0.623 + 0.781i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1048551655 + 0.2178954245i\)
\(L(\frac12)\) \(\approx\) \(0.1048551655 + 0.2178954245i\)
\(L(1)\) \(\approx\) \(0.7750344450 - 0.1177339012i\)
\(L(1)\) \(\approx\) \(0.7750344450 - 0.1177339012i\)

Euler product

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

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 $Z$-function along the critical line