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Properties

Label 2-34e2-68.15-c0-0-0
Degree $2$
Conductor $1156$
Sign $-0.957 - 0.287i$
Analytic cond. $0.576919$
Root an. cond. $0.759551$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 + 0.707i)2-s + 1.00i·4-s + (−1.30 + 0.541i)5-s + (−0.707 + 0.707i)8-s + (−0.707 + 0.707i)9-s + (−1.30 − 0.541i)10-s − 1.00·16-s − 1.00·18-s + (−0.541 − 1.30i)20-s + (0.707 − 0.707i)25-s + (−1.30 + 0.541i)29-s + (−0.707 − 0.707i)32-s + (−0.707 − 0.707i)36-s + (0.541 + 1.30i)37-s + (0.541 − 1.30i)40-s + (1.30 + 0.541i)41-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + 1.00i·4-s + (−1.30 + 0.541i)5-s + (−0.707 + 0.707i)8-s + (−0.707 + 0.707i)9-s + (−1.30 − 0.541i)10-s − 1.00·16-s − 1.00·18-s + (−0.541 − 1.30i)20-s + (0.707 − 0.707i)25-s + (−1.30 + 0.541i)29-s + (−0.707 − 0.707i)32-s + (−0.707 − 0.707i)36-s + (0.541 + 1.30i)37-s + (0.541 − 1.30i)40-s + (1.30 + 0.541i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 - 0.287i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 - 0.287i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1156\)    =    \(2^{2} \cdot 17^{2}\)
Sign: $-0.957 - 0.287i$
Analytic conductor: \(0.576919\)
Root analytic conductor: \(0.759551\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1156} (423, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1156,\ (\ :0),\ -0.957 - 0.287i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9076862289\)
\(L(\frac12)\) \(\approx\) \(0.9076862289\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
17 \( 1 \)
good3 \( 1 + (0.707 - 0.707i)T^{2} \)
5 \( 1 + (1.30 - 0.541i)T + (0.707 - 0.707i)T^{2} \)
7 \( 1 + (-0.707 - 0.707i)T^{2} \)
11 \( 1 + (0.707 + 0.707i)T^{2} \)
13 \( 1 - T^{2} \)
19 \( 1 - iT^{2} \)
23 \( 1 + (0.707 + 0.707i)T^{2} \)
29 \( 1 + (1.30 - 0.541i)T + (0.707 - 0.707i)T^{2} \)
31 \( 1 + (0.707 - 0.707i)T^{2} \)
37 \( 1 + (-0.541 - 1.30i)T + (-0.707 + 0.707i)T^{2} \)
41 \( 1 + (-1.30 - 0.541i)T + (0.707 + 0.707i)T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + (-1.30 - 0.541i)T + (0.707 + 0.707i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (0.707 - 0.707i)T^{2} \)
73 \( 1 + (-1.30 + 0.541i)T + (0.707 - 0.707i)T^{2} \)
79 \( 1 + (0.707 + 0.707i)T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-1.30 + 0.541i)T + (0.707 - 0.707i)T^{2} \)
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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

−10.68507051422760144121374816773, −9.291573296860272890560901922446, −8.332480244608848390602312511435, −7.77295741552102784138094403772, −7.16130951352636899773008498671, −6.18246281118323339340720698098, −5.25184694454619209279789249701, −4.31769717812614239343102746203, −3.46921444218388980636167900408, −2.56956287288719737630987979015, 0.63200268437517724297310985726, 2.39286927844528915535709674927, 3.68121970960635462551561965975, 4.04441639423184818876657611596, 5.21526756412162349981462282867, 5.98983432564125561071810254496, 7.10211969173743181882561708761, 8.042383277484357543948143696344, 8.978479629678215658197954846789, 9.565827559754162839605876211669

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