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Properties

Label 2-34e2-1156.511-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  + (0.0922 − 0.995i)2-s + (−0.982 − 0.183i)4-s + (0.149 + 0.526i)5-s + (−0.273 + 0.961i)8-s + (0.445 + 0.895i)9-s + (0.538 − 0.100i)10-s + (−0.243 + 0.857i)13-s + (0.932 + 0.361i)16-s + (0.932 + 0.361i)17-s + (0.932 − 0.361i)18-s + (−0.0505 − 0.544i)20-s + (0.595 − 0.368i)25-s + (0.831 + 0.322i)26-s + (−0.876 − 0.163i)29-s + (0.445 − 0.895i)32-s + ⋯
L(s)  = 1  + (0.0922 − 0.995i)2-s + (−0.982 − 0.183i)4-s + (0.149 + 0.526i)5-s + (−0.273 + 0.961i)8-s + (0.445 + 0.895i)9-s + (0.538 − 0.100i)10-s + (−0.243 + 0.857i)13-s + (0.932 + 0.361i)16-s + (0.932 + 0.361i)17-s + (0.932 − 0.361i)18-s + (−0.0505 − 0.544i)20-s + (0.595 − 0.368i)25-s + (0.831 + 0.322i)26-s + (−0.876 − 0.163i)29-s + (0.445 − 0.895i)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: $\chi_{1156} (511, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1156,\ (\ :0),\ 0.947 + 0.320i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.028272758\)
\(L(\frac12)\) \(\approx\) \(1.028272758\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0922 + 0.995i)T \)
17 \( 1 + (-0.932 - 0.361i)T \)
good3 \( 1 + (-0.445 - 0.895i)T^{2} \)
5 \( 1 + (-0.149 - 0.526i)T + (-0.850 + 0.526i)T^{2} \)
7 \( 1 + (0.602 - 0.798i)T^{2} \)
11 \( 1 + (-0.932 - 0.361i)T^{2} \)
13 \( 1 + (0.243 - 0.857i)T + (-0.850 - 0.526i)T^{2} \)
19 \( 1 + (0.982 - 0.183i)T^{2} \)
23 \( 1 + (0.602 - 0.798i)T^{2} \)
29 \( 1 + (0.876 + 0.163i)T + (0.932 + 0.361i)T^{2} \)
31 \( 1 + (0.850 + 0.526i)T^{2} \)
37 \( 1 + (-0.136 + 0.124i)T + (0.0922 - 0.995i)T^{2} \)
41 \( 1 + (-0.465 - 0.288i)T + (0.445 + 0.895i)T^{2} \)
43 \( 1 + (-0.739 + 0.673i)T^{2} \)
47 \( 1 + (0.602 + 0.798i)T^{2} \)
53 \( 1 + (0.537 + 1.07i)T + (-0.602 + 0.798i)T^{2} \)
59 \( 1 + (0.273 - 0.961i)T^{2} \)
61 \( 1 + (-1.02 + 1.35i)T + (-0.273 - 0.961i)T^{2} \)
67 \( 1 + (0.982 - 0.183i)T^{2} \)
71 \( 1 + (0.602 - 0.798i)T^{2} \)
73 \( 1 + (-0.831 - 0.322i)T + (0.739 + 0.673i)T^{2} \)
79 \( 1 + (0.982 - 0.183i)T^{2} \)
83 \( 1 + (-0.445 + 0.895i)T^{2} \)
89 \( 1 + (0.0505 + 0.177i)T + (-0.850 + 0.526i)T^{2} \)
97 \( 1 + (0.876 + 1.75i)T + (-0.602 + 0.798i)T^{2} \)
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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

−9.953477164719401041686363948857, −9.522795557698738440267417333839, −8.391470006701298987495516336703, −7.64281182875198289279389161523, −6.58692353749896977009609223009, −5.47634773724758441359996637950, −4.62252929534954542351558152884, −3.68261091018087324305168542351, −2.57070085652996480089114490547, −1.62158317312793800906900972005, 1.03519128124228648072767871759, 3.13913344029312502229628573526, 4.08483513147475854573868700891, 5.15945654741979196312564627919, 5.73739918944325260986483344859, 6.76799508878697614185479077450, 7.50681395052233405952337977921, 8.291335683936464543487307269437, 9.217450227575463275735028636677, 9.677530774044022371213489467797

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