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Properties

Label 2-34e2-1156.55-c0-0-0
Degree $2$
Conductor $1156$
Sign $0.947 + 0.320i$
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  + (−0.995 − 0.0922i)2-s + (0.982 + 0.183i)4-s + (0.850 + 0.473i)5-s + (−0.961 − 0.273i)8-s + (0.895 − 0.445i)9-s + (−0.802 − 0.549i)10-s + (0.489 − 1.72i)13-s + (0.932 + 0.361i)16-s + (−0.932 − 0.361i)17-s + (−0.932 + 0.361i)18-s + (0.748 + 0.621i)20-s + (−0.0278 − 0.0449i)25-s + (−0.646 + 1.66i)26-s + (0.524 − 0.359i)29-s + (−0.895 − 0.445i)32-s + ⋯
L(s)  = 1  + (−0.995 − 0.0922i)2-s + (0.982 + 0.183i)4-s + (0.850 + 0.473i)5-s + (−0.961 − 0.273i)8-s + (0.895 − 0.445i)9-s + (−0.802 − 0.549i)10-s + (0.489 − 1.72i)13-s + (0.932 + 0.361i)16-s + (−0.932 − 0.361i)17-s + (−0.932 + 0.361i)18-s + (0.748 + 0.621i)20-s + (−0.0278 − 0.0449i)25-s + (−0.646 + 1.66i)26-s + (0.524 − 0.359i)29-s + (−0.895 − 0.445i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1156\)    =    \(2^{2} \cdot 17^{2}\)
Sign: $0.947 + 0.320i$
Analytic conductor: \(0.576919\)
Root analytic conductor: \(0.759551\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1156} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1156,\ (\ :0),\ 0.947 + 0.320i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8424082521\)
\(L(\frac12)\) \(\approx\) \(0.8424082521\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.995 + 0.0922i)T \)
17 \( 1 + (0.932 + 0.361i)T \)
good3 \( 1 + (-0.895 + 0.445i)T^{2} \)
5 \( 1 + (-0.850 - 0.473i)T + (0.526 + 0.850i)T^{2} \)
7 \( 1 + (0.798 + 0.602i)T^{2} \)
11 \( 1 + (0.361 - 0.932i)T^{2} \)
13 \( 1 + (-0.489 + 1.72i)T + (-0.850 - 0.526i)T^{2} \)
19 \( 1 + (-0.982 + 0.183i)T^{2} \)
23 \( 1 + (0.798 + 0.602i)T^{2} \)
29 \( 1 + (-0.524 + 0.359i)T + (0.361 - 0.932i)T^{2} \)
31 \( 1 + (0.526 - 0.850i)T^{2} \)
37 \( 1 + (0.0710 - 1.53i)T + (-0.995 - 0.0922i)T^{2} \)
41 \( 1 + (-0.400 - 1.70i)T + (-0.895 + 0.445i)T^{2} \)
43 \( 1 + (0.739 - 0.673i)T^{2} \)
47 \( 1 + (0.602 + 0.798i)T^{2} \)
53 \( 1 + (1.42 - 0.711i)T + (0.602 - 0.798i)T^{2} \)
59 \( 1 + (-0.273 + 0.961i)T^{2} \)
61 \( 1 + (0.453 - 0.0632i)T + (0.961 - 0.273i)T^{2} \)
67 \( 1 + (0.982 - 0.183i)T^{2} \)
71 \( 1 + (-0.798 - 0.602i)T^{2} \)
73 \( 1 + (-1.73 + 0.765i)T + (0.673 - 0.739i)T^{2} \)
79 \( 1 + (0.183 + 0.982i)T^{2} \)
83 \( 1 + (0.445 - 0.895i)T^{2} \)
89 \( 1 + (-0.544 - 1.91i)T + (-0.850 + 0.526i)T^{2} \)
97 \( 1 + (-1.07 - 0.359i)T + (0.798 + 0.602i)T^{2} \)
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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

−9.920493163103735149947041347078, −9.368600745917698537023018453069, −8.285275844611064980011940793114, −7.67019906696150359595854707685, −6.44542248680303576618404441633, −6.29359334497639221571440656077, −4.90198159728431535173556900011, −3.38961461379273100149183096448, −2.50475855282238834839986922387, −1.20762991033269822769312722712, 1.58834062366811849672041022519, 2.15018172316033166030993071301, 3.92532765546713507026424850619, 5.00200542027922787642821138519, 6.12956679225965598007960464692, 6.77730588204890067517774034276, 7.58198232533170234183121573303, 8.706101594758543466009030138462, 9.187578601549798877984472354608, 9.806687674864215769781424792592

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