Properties

Label 2-34e2-1156.999-c0-0-0
Degree $2$
Conductor $1156$
Sign $0.624 + 0.781i$
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  + (−0.798 − 0.602i)2-s + (0.273 + 0.961i)4-s + (−0.739 − 0.326i)5-s + (0.361 − 0.932i)8-s + (−0.995 + 0.0922i)9-s + (0.393 + 0.705i)10-s + (1.85 + 0.719i)13-s + (−0.850 + 0.526i)16-s + (0.850 − 0.526i)17-s + (0.850 + 0.526i)18-s + (0.111 − 0.800i)20-s + (−0.234 − 0.256i)25-s + (−1.04 − 1.69i)26-s + (0.748 − 1.34i)29-s + (0.995 + 0.0922i)32-s + ⋯
L(s)  = 1  + (−0.798 − 0.602i)2-s + (0.273 + 0.961i)4-s + (−0.739 − 0.326i)5-s + (0.361 − 0.932i)8-s + (−0.995 + 0.0922i)9-s + (0.393 + 0.705i)10-s + (1.85 + 0.719i)13-s + (−0.850 + 0.526i)16-s + (0.850 − 0.526i)17-s + (0.850 + 0.526i)18-s + (0.111 − 0.800i)20-s + (−0.234 − 0.256i)25-s + (−1.04 − 1.69i)26-s + (0.748 − 1.34i)29-s + (0.995 + 0.0922i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.624 + 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.624 + 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1156\)    =    \(2^{2} \cdot 17^{2}\)
Sign: $0.624 + 0.781i$
Analytic conductor: \(0.576919\)
Root analytic conductor: \(0.759551\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1156} (999, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1156,\ (\ :0),\ 0.624 + 0.781i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6339641708\)
\(L(\frac12)\) \(\approx\) \(0.6339641708\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.798 + 0.602i)T \)
17 \( 1 + (-0.850 + 0.526i)T \)
good3 \( 1 + (0.995 - 0.0922i)T^{2} \)
5 \( 1 + (0.739 + 0.326i)T + (0.673 + 0.739i)T^{2} \)
7 \( 1 + (-0.183 - 0.982i)T^{2} \)
11 \( 1 + (-0.526 - 0.850i)T^{2} \)
13 \( 1 + (-1.85 - 0.719i)T + (0.739 + 0.673i)T^{2} \)
19 \( 1 + (-0.273 + 0.961i)T^{2} \)
23 \( 1 + (-0.183 - 0.982i)T^{2} \)
29 \( 1 + (-0.748 + 1.34i)T + (-0.526 - 0.850i)T^{2} \)
31 \( 1 + (0.673 - 0.739i)T^{2} \)
37 \( 1 + (-0.0878 + 0.261i)T + (-0.798 - 0.602i)T^{2} \)
41 \( 1 + (-1.82 + 0.0844i)T + (0.995 - 0.0922i)T^{2} \)
43 \( 1 + (0.445 + 0.895i)T^{2} \)
47 \( 1 + (0.982 + 0.183i)T^{2} \)
53 \( 1 + (0.365 - 0.0339i)T + (0.982 - 0.183i)T^{2} \)
59 \( 1 + (0.932 + 0.361i)T^{2} \)
61 \( 1 + (-1.12 + 1.64i)T + (-0.361 - 0.932i)T^{2} \)
67 \( 1 + (0.273 - 0.961i)T^{2} \)
71 \( 1 + (0.183 + 0.982i)T^{2} \)
73 \( 1 + (-1.24 - 0.292i)T + (0.895 + 0.445i)T^{2} \)
79 \( 1 + (-0.961 - 0.273i)T^{2} \)
83 \( 1 + (0.0922 - 0.995i)T^{2} \)
89 \( 1 + (1.48 - 0.576i)T + (0.739 - 0.673i)T^{2} \)
97 \( 1 + (1.11 - 1.34i)T + (-0.183 - 0.982i)T^{2} \)
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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

−9.726114792669753749951123057497, −9.066830204667653690111672252195, −8.170984036284369830007865730240, −7.952178372837805739437744112668, −6.65055715642403007885931433444, −5.78949430907393889040949356566, −4.30987277144579036828123708367, −3.59778504177246814750413860371, −2.50216767731928666820568373521, −0.957098284840174723029601516978, 1.16001575000915243741753499127, 2.93581227350271891311999678964, 3.87956989293361627405368327746, 5.41828177346288504166890234948, 5.96961350214367625648507248427, 6.87600688482695432594246717885, 7.888994622524675584832884243841, 8.367277093555646216971999039851, 9.002163258129608897045261947377, 10.11725682798887949674698349012

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