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Properties

Label 2-34e2-68.43-c0-0-2
Degree $2$
Conductor $1156$
Sign $0.0758 + 0.997i$
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 + (−0.707 − 0.707i)8-s + (0.707 + 0.707i)9-s − 2i·13-s − 1.00·16-s + 1.00·18-s + (0.707 + 0.707i)25-s + (−1.41 − 1.41i)26-s + (−0.707 + 0.707i)32-s + (0.707 − 0.707i)36-s + (−0.707 + 0.707i)49-s + 1.00·50-s − 2.00·52-s + (−1.41 + 1.41i)53-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s − 1.00i·4-s + (−0.707 − 0.707i)8-s + (0.707 + 0.707i)9-s − 2i·13-s − 1.00·16-s + 1.00·18-s + (0.707 + 0.707i)25-s + (−1.41 − 1.41i)26-s + (−0.707 + 0.707i)32-s + (0.707 − 0.707i)36-s + (−0.707 + 0.707i)49-s + 1.00·50-s − 2.00·52-s + (−1.41 + 1.41i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0758 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0758 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1156\)    =    \(2^{2} \cdot 17^{2}\)
Sign: $0.0758 + 0.997i$
Analytic conductor: \(0.576919\)
Root analytic conductor: \(0.759551\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1156} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1156,\ (\ :0),\ 0.0758 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.518752281\)
\(L(\frac12)\) \(\approx\) \(1.518752281\)
\(L(1)\) 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 \( 1 + (-0.707 - 0.707i)T^{2} \)
7 \( 1 + (0.707 - 0.707i)T^{2} \)
11 \( 1 + (-0.707 + 0.707i)T^{2} \)
13 \( 1 + 2iT - T^{2} \)
19 \( 1 + iT^{2} \)
23 \( 1 + (-0.707 + 0.707i)T^{2} \)
29 \( 1 + (-0.707 - 0.707i)T^{2} \)
31 \( 1 + (-0.707 - 0.707i)T^{2} \)
37 \( 1 + (0.707 + 0.707i)T^{2} \)
41 \( 1 + (-0.707 + 0.707i)T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (1.41 - 1.41i)T - iT^{2} \)
59 \( 1 - iT^{2} \)
61 \( 1 + (-0.707 + 0.707i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-0.707 - 0.707i)T^{2} \)
73 \( 1 + (-0.707 - 0.707i)T^{2} \)
79 \( 1 + (-0.707 + 0.707i)T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 - 2iT - T^{2} \)
97 \( 1 + (-0.707 - 0.707i)T^{2} \)
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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

−10.09258503253708068252045706764, −9.242018384954255314122508198696, −8.089526149872437346056415071278, −7.32473753966560610280228610445, −6.18874151828348691825630600809, −5.31161014283834064940063943495, −4.66354630754223888740597022685, −3.47817760586934465473963088615, −2.63818819517444431457608271640, −1.27108921017610214870066629693, 1.94000668090253920983587136554, 3.37186842420301803839305182207, 4.27584797284342223247705736725, 4.89797416652445988361001569566, 6.26544091862923101944558774158, 6.68481664619663487812150277937, 7.43770911518328827387415129516, 8.540129327357044549807926270715, 9.188386244521097214075755138820, 10.00903543783694252362392919977

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