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Properties

Label 2-34e2-1156.115-c0-0-0
Degree $2$
Conductor $1156$
Sign $0.252 + 0.967i$
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  + (−0.673 + 0.739i)2-s + (−0.0922 − 0.995i)4-s + (0.273 − 1.96i)5-s + (0.798 + 0.602i)8-s + (−0.526 − 0.850i)9-s + (1.26 + 1.52i)10-s + (−0.634 + 0.840i)13-s + (−0.982 + 0.183i)16-s + (0.982 − 0.183i)17-s + (0.982 + 0.183i)18-s + (−1.97 − 0.0914i)20-s + (−2.81 − 0.800i)25-s + (−0.193 − 1.03i)26-s + (0.292 − 0.352i)29-s + (0.526 − 0.850i)32-s + ⋯
L(s)  = 1  + (−0.673 + 0.739i)2-s + (−0.0922 − 0.995i)4-s + (0.273 − 1.96i)5-s + (0.798 + 0.602i)8-s + (−0.526 − 0.850i)9-s + (1.26 + 1.52i)10-s + (−0.634 + 0.840i)13-s + (−0.982 + 0.183i)16-s + (0.982 − 0.183i)17-s + (0.982 + 0.183i)18-s + (−1.97 − 0.0914i)20-s + (−2.81 − 0.800i)25-s + (−0.193 − 1.03i)26-s + (0.292 − 0.352i)29-s + (0.526 − 0.850i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.252 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.252 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1156\)    =    \(2^{2} \cdot 17^{2}\)
Sign: $0.252 + 0.967i$
Analytic conductor: \(0.576919\)
Root analytic conductor: \(0.759551\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1156} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1156,\ (\ :0),\ 0.252 + 0.967i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6603338808\)
\(L(\frac12)\) \(\approx\) \(0.6603338808\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.673 - 0.739i)T \)
17 \( 1 + (-0.982 + 0.183i)T \)
good3 \( 1 + (0.526 + 0.850i)T^{2} \)
5 \( 1 + (-0.273 + 1.96i)T + (-0.961 - 0.273i)T^{2} \)
7 \( 1 + (0.895 + 0.445i)T^{2} \)
11 \( 1 + (-0.183 - 0.982i)T^{2} \)
13 \( 1 + (0.634 - 0.840i)T + (-0.273 - 0.961i)T^{2} \)
19 \( 1 + (0.0922 - 0.995i)T^{2} \)
23 \( 1 + (0.895 + 0.445i)T^{2} \)
29 \( 1 + (-0.292 + 0.352i)T + (-0.183 - 0.982i)T^{2} \)
31 \( 1 + (-0.961 + 0.273i)T^{2} \)
37 \( 1 + (0.806 + 1.82i)T + (-0.673 + 0.739i)T^{2} \)
41 \( 1 + (0.241 + 0.134i)T + (0.526 + 0.850i)T^{2} \)
43 \( 1 + (0.932 + 0.361i)T^{2} \)
47 \( 1 + (-0.445 - 0.895i)T^{2} \)
53 \( 1 + (-0.942 - 1.52i)T + (-0.445 + 0.895i)T^{2} \)
59 \( 1 + (-0.602 + 0.798i)T^{2} \)
61 \( 1 + (0.555 + 1.65i)T + (-0.798 + 0.602i)T^{2} \)
67 \( 1 + (-0.0922 + 0.995i)T^{2} \)
71 \( 1 + (-0.895 - 0.445i)T^{2} \)
73 \( 1 + (-1.60 - 1.10i)T + (0.361 + 0.932i)T^{2} \)
79 \( 1 + (0.995 + 0.0922i)T^{2} \)
83 \( 1 + (-0.850 - 0.526i)T^{2} \)
89 \( 1 + (-0.811 - 1.07i)T + (-0.273 + 0.961i)T^{2} \)
97 \( 1 + (-1.49 - 0.352i)T + (0.895 + 0.445i)T^{2} \)
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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

−9.333688244326761326538963502742, −9.154051763479939062460939356974, −8.313281876677314345836600078995, −7.55530688971962425287670878131, −6.43182582564870716859421188205, −5.56205890388131991536388953566, −4.98839309598117566680630451865, −3.97225669182317721740418469913, −1.96331692636842518958448474121, −0.73585452035323312860481590062, 1.98349781007112786245706999438, 2.94869300572194261368293591402, 3.39312565642507451973618422417, 5.00941652262899186696422606903, 6.15066736687414422899452201148, 7.12592306954322772502999502498, 7.73010265921788025298865132575, 8.452219634610472014493225375633, 9.819533013211383295623262562708, 10.21684049084134261325071065347

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