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Properties

Label 2-1133-1133.538-c0-0-0
Degree $2$
Conductor $1133$
Sign $0.492 + 0.870i$
Analytic cond. $0.565440$
Root an. cond. $0.751957$
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  + (−1.25 − 0.778i)3-s + (−0.602 + 0.798i)4-s + (−0.404 − 0.368i)5-s + (0.527 + 1.06i)9-s + (0.445 + 0.895i)11-s + (1.37 − 0.533i)12-s + (0.221 + 0.778i)15-s + (−0.273 − 0.961i)16-s + (0.538 − 0.100i)20-s + (0.658 − 1.32i)23-s + (−0.0646 − 0.697i)25-s + (0.0251 − 0.271i)27-s + (−0.404 − 1.42i)31-s + (0.136 − 1.47i)33-s + (−1.16 − 0.217i)36-s + (1.73 − 0.673i)37-s + ⋯
L(s)  = 1  + (−1.25 − 0.778i)3-s + (−0.602 + 0.798i)4-s + (−0.404 − 0.368i)5-s + (0.527 + 1.06i)9-s + (0.445 + 0.895i)11-s + (1.37 − 0.533i)12-s + (0.221 + 0.778i)15-s + (−0.273 − 0.961i)16-s + (0.538 − 0.100i)20-s + (0.658 − 1.32i)23-s + (−0.0646 − 0.697i)25-s + (0.0251 − 0.271i)27-s + (−0.404 − 1.42i)31-s + (0.136 − 1.47i)33-s + (−1.16 − 0.217i)36-s + (1.73 − 0.673i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1133 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.492 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1133 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.492 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1133\)    =    \(11 \cdot 103\)
Sign: $0.492 + 0.870i$
Analytic conductor: \(0.565440\)
Root analytic conductor: \(0.751957\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1133} (538, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1133,\ (\ :0),\ 0.492 + 0.870i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5005753471\)
\(L(\frac12)\) \(\approx\) \(0.5005753471\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-0.445 - 0.895i)T \)
103 \( 1 + (0.273 + 0.961i)T \)
good2 \( 1 + (0.602 - 0.798i)T^{2} \)
3 \( 1 + (1.25 + 0.778i)T + (0.445 + 0.895i)T^{2} \)
5 \( 1 + (0.404 + 0.368i)T + (0.0922 + 0.995i)T^{2} \)
7 \( 1 + (-0.932 + 0.361i)T^{2} \)
13 \( 1 + (-0.932 + 0.361i)T^{2} \)
17 \( 1 + (0.982 - 0.183i)T^{2} \)
19 \( 1 + (-0.445 + 0.895i)T^{2} \)
23 \( 1 + (-0.658 + 1.32i)T + (-0.602 - 0.798i)T^{2} \)
29 \( 1 + (-0.0922 - 0.995i)T^{2} \)
31 \( 1 + (0.404 + 1.42i)T + (-0.850 + 0.526i)T^{2} \)
37 \( 1 + (-1.73 + 0.673i)T + (0.739 - 0.673i)T^{2} \)
41 \( 1 + (-0.0922 + 0.995i)T^{2} \)
43 \( 1 + (-0.739 - 0.673i)T^{2} \)
47 \( 1 - 0.184T + T^{2} \)
53 \( 1 + (-1.02 + 0.634i)T + (0.445 - 0.895i)T^{2} \)
59 \( 1 + (1.83 + 0.342i)T + (0.932 + 0.361i)T^{2} \)
61 \( 1 + (0.982 - 0.183i)T^{2} \)
67 \( 1 + (-1.93 - 0.361i)T + (0.932 + 0.361i)T^{2} \)
71 \( 1 + (1.25 - 1.14i)T + (0.0922 - 0.995i)T^{2} \)
73 \( 1 + (-0.0922 + 0.995i)T^{2} \)
79 \( 1 + (-0.0922 - 0.995i)T^{2} \)
83 \( 1 + (-0.932 + 0.361i)T^{2} \)
89 \( 1 + (-0.329 - 0.436i)T + (-0.273 + 0.961i)T^{2} \)
97 \( 1 + (0.0505 - 0.544i)T + (-0.982 - 0.183i)T^{2} \)
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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

−9.844339532150819656281316849843, −8.987687778989192882983397578186, −8.076164318824256979213051199627, −7.34051082646060279675239302875, −6.63665379063008038893828601931, −5.65497475247581896789685923152, −4.62244430226771971698934626167, −4.06384468422814164011687173725, −2.40358524789628570564432822177, −0.67588497555393846784253545750, 1.12904371337099903586090585369, 3.32119071815744527040880380562, 4.25147395836222016142356015525, 5.13209003647446678827502383715, 5.77570438512143406703722522240, 6.47894424666057772738265050458, 7.58695483760776509702072486980, 8.864767611375271891181645020862, 9.444384922136988187832077819589, 10.32198138967304142778747892378

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