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Properties

Label 2-34e2-17.15-c1-0-3
Degree $2$
Conductor $1156$
Sign $-0.168 - 0.985i$
Analytic cond. $9.23070$
Root an. cond. $3.03820$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.292 + 0.707i)3-s + (−0.707 + 0.292i)5-s + (−1.70 − 0.707i)7-s + (1.70 − 1.70i)9-s + (−1.12 + 2.70i)11-s + 1.17i·13-s + (−0.414 − 0.414i)15-s + (3.82 + 3.82i)19-s − 1.41i·21-s + (0.535 − 1.29i)23-s + (−3.12 + 3.12i)25-s + (3.82 + 1.58i)27-s + (−5.53 + 2.29i)29-s + (3.94 + 9.53i)31-s − 2.24·33-s + ⋯
L(s)  = 1  + (0.169 + 0.408i)3-s + (−0.316 + 0.130i)5-s + (−0.645 − 0.267i)7-s + (0.569 − 0.569i)9-s + (−0.338 + 0.816i)11-s + 0.324i·13-s + (−0.106 − 0.106i)15-s + (0.878 + 0.878i)19-s − 0.308i·21-s + (0.111 − 0.269i)23-s + (−0.624 + 0.624i)25-s + (0.736 + 0.305i)27-s + (−1.02 + 0.425i)29-s + (0.709 + 1.71i)31-s − 0.390·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.168 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.168 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1156\)    =    \(2^{2} \cdot 17^{2}\)
Sign: $-0.168 - 0.985i$
Analytic conductor: \(9.23070\)
Root analytic conductor: \(3.03820\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1156} (1001, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1156,\ (\ :1/2),\ -0.168 - 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.234709805\)
\(L(\frac12)\) \(\approx\) \(1.234709805\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 \)
good3 \( 1 + (-0.292 - 0.707i)T + (-2.12 + 2.12i)T^{2} \)
5 \( 1 + (0.707 - 0.292i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (1.70 + 0.707i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (1.12 - 2.70i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 - 1.17iT - 13T^{2} \)
19 \( 1 + (-3.82 - 3.82i)T + 19iT^{2} \)
23 \( 1 + (-0.535 + 1.29i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (5.53 - 2.29i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + (-3.94 - 9.53i)T + (-21.9 + 21.9i)T^{2} \)
37 \( 1 + (-1.87 - 4.53i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (7.53 + 3.12i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (-3 + 3i)T - 43iT^{2} \)
47 \( 1 - 7.65iT - 47T^{2} \)
53 \( 1 + (-9.82 - 9.82i)T + 53iT^{2} \)
59 \( 1 + (-4.17 + 4.17i)T - 59iT^{2} \)
61 \( 1 + (0.707 + 0.292i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + 11.3T + 67T^{2} \)
71 \( 1 + (-3.12 - 7.53i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (4.70 - 1.94i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (3.46 - 8.36i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (5.82 + 5.82i)T + 83iT^{2} \)
89 \( 1 + 5.17iT - 89T^{2} \)
97 \( 1 + (12.3 - 5.12i)T + (68.5 - 68.5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.988661841911018792208639217431, −9.389055670177774766944108086968, −8.478992207256951721879972119674, −7.30547777041216466849480688863, −6.96489382442785167464942746659, −5.78850667379390005389453128614, −4.71209616800838438125353025189, −3.81405264293460123731197426421, −3.07446116846016556189355284672, −1.46425238485708285966436335177, 0.54507525774588987198239038861, 2.20014642751106620357389178940, 3.19491335932151178049542690787, 4.28434068437021909115262298450, 5.40370827595610317572028777335, 6.18936792108327755057227268505, 7.26276494559398209320894538677, 7.84384670601509052039279148830, 8.648520599163983195600101419792, 9.618005241850250901010066382092

Graph of the $Z$-function along the critical line