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Properties

Label 2-1133-1133.76-c0-0-0
Degree $2$
Conductor $1133$
Sign $0.916 + 0.400i$
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  + (0.658 + 0.600i)3-s + (−0.982 + 0.183i)4-s + (0.831 − 1.66i)5-s + (−0.0189 − 0.204i)9-s + (0.0922 + 0.995i)11-s + (−0.757 − 0.469i)12-s + (1.55 − 0.600i)15-s + (0.932 − 0.361i)16-s + (−0.510 + 1.79i)20-s + (0.0822 − 0.887i)23-s + (−1.49 − 1.97i)25-s + (0.647 − 0.857i)27-s + (0.831 − 0.322i)31-s + (−0.537 + 0.711i)33-s + (0.0561 + 0.197i)36-s + (1.44 + 0.895i)37-s + ⋯
L(s)  = 1  + (0.658 + 0.600i)3-s + (−0.982 + 0.183i)4-s + (0.831 − 1.66i)5-s + (−0.0189 − 0.204i)9-s + (0.0922 + 0.995i)11-s + (−0.757 − 0.469i)12-s + (1.55 − 0.600i)15-s + (0.932 − 0.361i)16-s + (−0.510 + 1.79i)20-s + (0.0822 − 0.887i)23-s + (−1.49 − 1.97i)25-s + (0.647 − 0.857i)27-s + (0.831 − 0.322i)31-s + (−0.537 + 0.711i)33-s + (0.0561 + 0.197i)36-s + (1.44 + 0.895i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1133 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 + 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1133 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 + 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1133\)    =    \(11 \cdot 103\)
Sign: $0.916 + 0.400i$
Analytic conductor: \(0.565440\)
Root analytic conductor: \(0.751957\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1133} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1133,\ (\ :0),\ 0.916 + 0.400i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.195561786\)
\(L(\frac12)\) \(\approx\) \(1.195561786\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-0.0922 - 0.995i)T \)
103 \( 1 + (-0.932 + 0.361i)T \)
good2 \( 1 + (0.982 - 0.183i)T^{2} \)
3 \( 1 + (-0.658 - 0.600i)T + (0.0922 + 0.995i)T^{2} \)
5 \( 1 + (-0.831 + 1.66i)T + (-0.602 - 0.798i)T^{2} \)
7 \( 1 + (0.850 + 0.526i)T^{2} \)
13 \( 1 + (0.850 + 0.526i)T^{2} \)
17 \( 1 + (0.273 - 0.961i)T^{2} \)
19 \( 1 + (-0.0922 + 0.995i)T^{2} \)
23 \( 1 + (-0.0822 + 0.887i)T + (-0.982 - 0.183i)T^{2} \)
29 \( 1 + (0.602 + 0.798i)T^{2} \)
31 \( 1 + (-0.831 + 0.322i)T + (0.739 - 0.673i)T^{2} \)
37 \( 1 + (-1.44 - 0.895i)T + (0.445 + 0.895i)T^{2} \)
41 \( 1 + (0.602 - 0.798i)T^{2} \)
43 \( 1 + (-0.445 + 0.895i)T^{2} \)
47 \( 1 + 1.20T + T^{2} \)
53 \( 1 + (1.45 - 1.32i)T + (0.0922 - 0.995i)T^{2} \)
59 \( 1 + (-0.465 - 1.63i)T + (-0.850 + 0.526i)T^{2} \)
61 \( 1 + (0.273 - 0.961i)T^{2} \)
67 \( 1 + (-0.149 - 0.526i)T + (-0.850 + 0.526i)T^{2} \)
71 \( 1 + (-0.658 - 1.32i)T + (-0.602 + 0.798i)T^{2} \)
73 \( 1 + (0.602 - 0.798i)T^{2} \)
79 \( 1 + (0.602 + 0.798i)T^{2} \)
83 \( 1 + (0.850 + 0.526i)T^{2} \)
89 \( 1 + (1.83 + 0.342i)T + (0.932 + 0.361i)T^{2} \)
97 \( 1 + (1.12 - 1.48i)T + (-0.273 - 0.961i)T^{2} \)
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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

−9.651993476616102730665155570459, −9.299782058872353734579183245778, −8.464412509951651914141168466434, −8.051379166332461656411204071577, −6.46227168849612434401836380263, −5.41458611181931135525231569544, −4.46395064934533306264904720533, −4.30208714913435571038936233645, −2.74169788216972062732799180886, −1.21046915722304125392600392807, 1.71242927916588460958080565324, 2.89826973384239265415046701875, 3.53604542859081792087635894612, 5.05675867007446436871779597639, 6.01511921675545201243761867366, 6.67922942396274381118924641643, 7.75072827415187609956102879826, 8.291126544862596009313921371747, 9.431334160643054898601972532631, 9.848478936491220400583567107745

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