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Properties

Label 2-34e2-17.8-c1-0-0
Degree $2$
Conductor $1156$
Sign $-0.440 - 0.897i$
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  + (0.280 − 0.676i)3-s + (−3.20 − 1.32i)5-s + (−0.676 + 0.280i)7-s + (1.74 + 1.74i)9-s + (−1.81 − 4.37i)11-s − 1.46i·13-s + (−1.79 + 1.79i)15-s + (−3.86 + 3.86i)19-s + 0.535i·21-s + (1.81 + 4.37i)23-s + (4.94 + 4.94i)25-s + (3.69 − 1.53i)27-s + (−3.20 − 1.32i)29-s + (−2.37 + 5.72i)31-s − 3.46·33-s + ⋯
L(s)  = 1  + (0.161 − 0.390i)3-s + (−1.43 − 0.592i)5-s + (−0.255 + 0.105i)7-s + (0.580 + 0.580i)9-s + (−0.546 − 1.31i)11-s − 0.406i·13-s + (−0.462 + 0.462i)15-s + (−0.886 + 0.886i)19-s + 0.116i·21-s + (0.377 + 0.911i)23-s + (0.989 + 0.989i)25-s + (0.711 − 0.294i)27-s + (−0.594 − 0.246i)29-s + (−0.425 + 1.02i)31-s − 0.603·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.440 - 0.897i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.440 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1156\)    =    \(2^{2} \cdot 17^{2}\)
Sign: $-0.440 - 0.897i$
Analytic conductor: \(9.23070\)
Root analytic conductor: \(3.03820\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1156} (977, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1156,\ (\ :1/2),\ -0.440 - 0.897i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2529515651\)
\(L(\frac12)\) \(\approx\) \(0.2529515651\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.280 + 0.676i)T + (-2.12 - 2.12i)T^{2} \)
5 \( 1 + (3.20 + 1.32i)T + (3.53 + 3.53i)T^{2} \)
7 \( 1 + (0.676 - 0.280i)T + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (1.81 + 4.37i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + 1.46iT - 13T^{2} \)
19 \( 1 + (3.86 - 3.86i)T - 19iT^{2} \)
23 \( 1 + (-1.81 - 4.37i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (3.20 + 1.32i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + (2.37 - 5.72i)T + (-21.9 - 21.9i)T^{2} \)
37 \( 1 + (4.38 - 10.5i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (-5.54 + 2.29i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (8.76 + 8.76i)T + 43iT^{2} \)
47 \( 1 - 6.92iT - 47T^{2} \)
53 \( 1 + (-0.656 + 0.656i)T - 53iT^{2} \)
59 \( 1 + (-6.69 - 6.69i)T + 59iT^{2} \)
61 \( 1 + (6.89 - 2.85i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + 1.07T + 67T^{2} \)
71 \( 1 + (0.840 - 2.02i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (1.84 + 0.765i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (0.690 + 1.66i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (6.69 - 6.69i)T - 83iT^{2} \)
89 \( 1 + 9.46iT - 89T^{2} \)
97 \( 1 + (-8.24 - 3.41i)T + (68.5 + 68.5i)T^{2} \)
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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

−10.22507469837919702497884905763, −8.904893317160006217313835410581, −8.272300816467846419604003804290, −7.79029175698306651053917984583, −6.96459229379234755280129481111, −5.77587648004534393326552609651, −4.87491799889109125728233744170, −3.84045737514724276833344470683, −3.03936481628125170562402657532, −1.38577023391561037788757116311, 0.11150720338223762920252259652, 2.25148620542839419578021677713, 3.47528722522059890035460081377, 4.20942320736809902987687620204, 4.85992622163929946830302790724, 6.52069504285811672614891149277, 7.09406752630823250440636890145, 7.72204190988014418521062928617, 8.762073111162866020038537881937, 9.563501740062191165549662579824

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