Properties

Label 2-1133-1133.549-c0-0-0
Degree $2$
Conductor $1133$
Sign $-0.750 + 0.661i$
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  + (−0.510 + 1.79i)3-s + (0.445 + 0.895i)4-s + (−1.12 − 0.435i)5-s + (−2.10 − 1.30i)9-s + (−0.850 − 0.526i)11-s + (−1.83 + 0.342i)12-s + (1.35 − 1.79i)15-s + (−0.602 + 0.798i)16-s + (−0.111 − 1.20i)20-s + (−1.58 + 0.981i)23-s + (0.334 + 0.304i)25-s + (2.03 − 1.85i)27-s + (−1.12 + 1.48i)31-s + (1.37 − 1.25i)33-s + (0.228 − 2.46i)36-s + (1.93 − 0.361i)37-s + ⋯
L(s)  = 1  + (−0.510 + 1.79i)3-s + (0.445 + 0.895i)4-s + (−1.12 − 0.435i)5-s + (−2.10 − 1.30i)9-s + (−0.850 − 0.526i)11-s + (−1.83 + 0.342i)12-s + (1.35 − 1.79i)15-s + (−0.602 + 0.798i)16-s + (−0.111 − 1.20i)20-s + (−1.58 + 0.981i)23-s + (0.334 + 0.304i)25-s + (2.03 − 1.85i)27-s + (−1.12 + 1.48i)31-s + (1.37 − 1.25i)33-s + (0.228 − 2.46i)36-s + (1.93 − 0.361i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1133 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.750 + 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1133 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.750 + 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1133\)    =    \(11 \cdot 103\)
Sign: $-0.750 + 0.661i$
Analytic conductor: \(0.565440\)
Root analytic conductor: \(0.751957\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1133} (549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1133,\ (\ :0),\ -0.750 + 0.661i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3664963366\)
\(L(\frac12)\) \(\approx\) \(0.3664963366\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (0.850 + 0.526i)T \)
103 \( 1 + (0.602 - 0.798i)T \)
good2 \( 1 + (-0.445 - 0.895i)T^{2} \)
3 \( 1 + (0.510 - 1.79i)T + (-0.850 - 0.526i)T^{2} \)
5 \( 1 + (1.12 + 0.435i)T + (0.739 + 0.673i)T^{2} \)
7 \( 1 + (0.982 - 0.183i)T^{2} \)
13 \( 1 + (0.982 - 0.183i)T^{2} \)
17 \( 1 + (-0.0922 - 0.995i)T^{2} \)
19 \( 1 + (0.850 - 0.526i)T^{2} \)
23 \( 1 + (1.58 - 0.981i)T + (0.445 - 0.895i)T^{2} \)
29 \( 1 + (-0.739 - 0.673i)T^{2} \)
31 \( 1 + (1.12 - 1.48i)T + (-0.273 - 0.961i)T^{2} \)
37 \( 1 + (-1.93 + 0.361i)T + (0.932 - 0.361i)T^{2} \)
41 \( 1 + (-0.739 + 0.673i)T^{2} \)
43 \( 1 + (-0.932 - 0.361i)T^{2} \)
47 \( 1 - 1.47T + T^{2} \)
53 \( 1 + (0.243 + 0.857i)T + (-0.850 + 0.526i)T^{2} \)
59 \( 1 + (0.181 - 1.95i)T + (-0.982 - 0.183i)T^{2} \)
61 \( 1 + (-0.0922 - 0.995i)T^{2} \)
67 \( 1 + (-0.0170 + 0.183i)T + (-0.982 - 0.183i)T^{2} \)
71 \( 1 + (0.510 - 0.197i)T + (0.739 - 0.673i)T^{2} \)
73 \( 1 + (-0.739 + 0.673i)T^{2} \)
79 \( 1 + (-0.739 - 0.673i)T^{2} \)
83 \( 1 + (0.982 - 0.183i)T^{2} \)
89 \( 1 + (0.537 - 1.07i)T + (-0.602 - 0.798i)T^{2} \)
97 \( 1 + (0.890 - 0.811i)T + (0.0922 - 0.995i)T^{2} \)
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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

−10.72815584677180160330366357920, −9.753278646736189450405036712020, −8.864534672219203130567909999054, −8.169180214207408864353987775031, −7.48784852624090746428006654462, −6.07123766411401349528725763854, −5.24507621592208067688201892871, −4.19753374117526743882254944423, −3.77816702566480655936988210014, −2.84791779581201575905975620708, 0.32535972346002006726595005333, 1.90001637938250886845364645866, 2.69170092088373534817663560400, 4.40133157667709688177149355682, 5.65605143627653294810799548886, 6.22563980262735259991175811444, 7.08384084245605028236894575496, 7.70814667126997986631727782259, 8.141574803907230317705220072678, 9.632870813262841198908321892695

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