Properties

Label 2-3808-3808.3093-c0-0-5
Degree 22
Conductor 38083808
Sign 0.831+0.555i0.831 + 0.555i
Analytic cond. 1.900431.90043
Root an. cond. 1.378561.37856
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  + (−0.891 − 0.453i)2-s + (1.84 − 0.763i)3-s + (0.587 + 0.809i)4-s + (−0.399 + 0.965i)5-s + (−1.98 − 0.156i)6-s + (0.707 + 0.707i)7-s + (−0.156 − 0.987i)8-s + (2.10 − 2.10i)9-s + (0.794 − 0.678i)10-s + (1.70 + 1.04i)12-s + (−0.309 − 0.951i)14-s + 2.08i·15-s + (−0.309 + 0.951i)16-s i·17-s + (−2.82 + 0.919i)18-s + ⋯
L(s)  = 1  + (−0.891 − 0.453i)2-s + (1.84 − 0.763i)3-s + (0.587 + 0.809i)4-s + (−0.399 + 0.965i)5-s + (−1.98 − 0.156i)6-s + (0.707 + 0.707i)7-s + (−0.156 − 0.987i)8-s + (2.10 − 2.10i)9-s + (0.794 − 0.678i)10-s + (1.70 + 1.04i)12-s + (−0.309 − 0.951i)14-s + 2.08i·15-s + (−0.309 + 0.951i)16-s i·17-s + (−2.82 + 0.919i)18-s + ⋯

Functional equation

Λ(s)=(3808s/2ΓC(s)L(s)=((0.831+0.555i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 3808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.831 + 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3808s/2ΓC(s)L(s)=((0.831+0.555i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 3808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.831 + 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 38083808    =    257172^{5} \cdot 7 \cdot 17
Sign: 0.831+0.555i0.831 + 0.555i
Analytic conductor: 1.900431.90043
Root analytic conductor: 1.378561.37856
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3808(3093,)\chi_{3808} (3093, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3808, ( :0), 0.831+0.555i)(2,\ 3808,\ (\ :0),\ 0.831 + 0.555i)

Particular Values

L(12)L(\frac{1}{2}) \approx 1.7351825331.735182533
L(12)L(\frac12) \approx 1.7351825331.735182533
L(1)L(1) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(0.891+0.453i)T 1 + (0.891 + 0.453i)T
7 1+(0.7070.707i)T 1 + (-0.707 - 0.707i)T
17 1+iT 1 + iT
good3 1+(1.84+0.763i)T+(0.7070.707i)T2 1 + (-1.84 + 0.763i)T + (0.707 - 0.707i)T^{2}
5 1+(0.3990.965i)T+(0.7070.707i)T2 1 + (0.399 - 0.965i)T + (-0.707 - 0.707i)T^{2}
11 1+(0.7070.707i)T2 1 + (-0.707 - 0.707i)T^{2}
13 1+(0.7070.707i)T2 1 + (0.707 - 0.707i)T^{2}
19 1+(0.7070.707i)T2 1 + (0.707 - 0.707i)T^{2}
23 1+iT2 1 + iT^{2}
29 1+(0.707+0.707i)T2 1 + (-0.707 + 0.707i)T^{2}
31 1+1.61T+T2 1 + 1.61T + T^{2}
37 1+(0.707+0.707i)T2 1 + (0.707 + 0.707i)T^{2}
41 1+(0.437+0.437i)TiT2 1 + (-0.437 + 0.437i)T - iT^{2}
43 1+(1.400.581i)T+(0.707+0.707i)T2 1 + (-1.40 - 0.581i)T + (0.707 + 0.707i)T^{2}
47 1+T2 1 + T^{2}
53 1+(0.431+0.178i)T+(0.707+0.707i)T2 1 + (0.431 + 0.178i)T + (0.707 + 0.707i)T^{2}
59 1+(0.707+0.707i)T2 1 + (0.707 + 0.707i)T^{2}
61 1+(0.144+0.0600i)T+(0.7070.707i)T2 1 + (-0.144 + 0.0600i)T + (0.707 - 0.707i)T^{2}
67 1+(1.790.744i)T+(0.7070.707i)T2 1 + (1.79 - 0.744i)T + (0.707 - 0.707i)T^{2}
71 1iT2 1 - iT^{2}
73 1+(1.261.26i)TiT2 1 + (1.26 - 1.26i)T - iT^{2}
79 1+T2 1 + T^{2}
83 1+(0.7070.707i)T2 1 + (0.707 - 0.707i)T^{2}
89 1iT2 1 - iT^{2}
97 1+0.312T+T2 1 + 0.312T + T^{2}
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   L(s)=p j=12(1αj,pps)1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−8.656018247697836241172255719718, −7.913369755834157983232969404453, −7.28544362525349604902473821146, −7.11649380365519185913604281375, −5.98477640933943450798636176260, −4.34905309751552634578316664804, −3.46416038552818959516292458513, −2.79252869485261193126758995328, −2.28068133989515754350352580782, −1.33681695998981928058494434061, 1.36665105871206143372463233264, 2.08333300721499710021826087433, 3.30282161001141153782728478825, 4.24670188848403994767315032047, 4.68589039847777592632955905059, 5.69124173671094522658253436976, 7.09938866695727644459531909972, 7.68630105563150734356371123895, 8.096247387074179449215996984122, 8.797203119174070195323187953777

Graph of the ZZ-function along the critical line