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Properties

Label 2-1133-1133.802-c0-0-0
Degree $2$
Conductor $1133$
Sign $-0.447 - 0.894i$
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  + (0.0822 − 0.165i)3-s + (−0.273 + 0.961i)4-s + (−0.156 + 1.69i)5-s + (0.582 + 0.770i)9-s + (−0.602 − 0.798i)11-s + (0.136 + 0.124i)12-s + (0.266 + 0.165i)15-s + (−0.850 − 0.526i)16-s + (−1.58 − 0.614i)20-s + (−0.111 + 0.147i)23-s + (−1.85 − 0.347i)25-s + (0.356 − 0.0666i)27-s + (−0.156 − 0.0971i)31-s + (−0.181 + 0.0339i)33-s + (−0.900 + 0.348i)36-s + (1.09 + 0.995i)37-s + ⋯
L(s)  = 1  + (0.0822 − 0.165i)3-s + (−0.273 + 0.961i)4-s + (−0.156 + 1.69i)5-s + (0.582 + 0.770i)9-s + (−0.602 − 0.798i)11-s + (0.136 + 0.124i)12-s + (0.266 + 0.165i)15-s + (−0.850 − 0.526i)16-s + (−1.58 − 0.614i)20-s + (−0.111 + 0.147i)23-s + (−1.85 − 0.347i)25-s + (0.356 − 0.0666i)27-s + (−0.156 − 0.0971i)31-s + (−0.181 + 0.0339i)33-s + (−0.900 + 0.348i)36-s + (1.09 + 0.995i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1133 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1133 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1133\)    =    \(11 \cdot 103\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(0.565440\)
Root analytic conductor: \(0.751957\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1133} (802, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1133,\ (\ :0),\ -0.447 - 0.894i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9065208538\)
\(L(\frac12)\) \(\approx\) \(0.9065208538\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (0.602 + 0.798i)T \)
103 \( 1 + (0.850 + 0.526i)T \)
good2 \( 1 + (0.273 - 0.961i)T^{2} \)
3 \( 1 + (-0.0822 + 0.165i)T + (-0.602 - 0.798i)T^{2} \)
5 \( 1 + (0.156 - 1.69i)T + (-0.982 - 0.183i)T^{2} \)
7 \( 1 + (-0.739 - 0.673i)T^{2} \)
13 \( 1 + (-0.739 - 0.673i)T^{2} \)
17 \( 1 + (-0.932 - 0.361i)T^{2} \)
19 \( 1 + (0.602 - 0.798i)T^{2} \)
23 \( 1 + (0.111 - 0.147i)T + (-0.273 - 0.961i)T^{2} \)
29 \( 1 + (0.982 + 0.183i)T^{2} \)
31 \( 1 + (0.156 + 0.0971i)T + (0.445 + 0.895i)T^{2} \)
37 \( 1 + (-1.09 - 0.995i)T + (0.0922 + 0.995i)T^{2} \)
41 \( 1 + (0.982 - 0.183i)T^{2} \)
43 \( 1 + (-0.0922 + 0.995i)T^{2} \)
47 \( 1 + 1.96T + T^{2} \)
53 \( 1 + (0.243 + 0.489i)T + (-0.602 + 0.798i)T^{2} \)
59 \( 1 + (-1.37 + 0.533i)T + (0.739 - 0.673i)T^{2} \)
61 \( 1 + (-0.932 - 0.361i)T^{2} \)
67 \( 1 + (-1.73 + 0.673i)T + (0.739 - 0.673i)T^{2} \)
71 \( 1 + (-0.0822 - 0.887i)T + (-0.982 + 0.183i)T^{2} \)
73 \( 1 + (0.982 - 0.183i)T^{2} \)
79 \( 1 + (0.982 + 0.183i)T^{2} \)
83 \( 1 + (-0.739 - 0.673i)T^{2} \)
89 \( 1 + (-0.465 - 1.63i)T + (-0.850 + 0.526i)T^{2} \)
97 \( 1 + (-1.67 + 0.312i)T + (0.932 - 0.361i)T^{2} \)
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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

−10.33653841262862299023786147122, −9.607120058662331446221079384831, −8.235414400290370737681832986214, −7.890192325359134869509435342977, −7.05551076378200879812972789693, −6.38587694230658543767745441570, −5.09565509813729448184397585825, −3.91113376364858386939874774220, −3.08427219119491810505205370232, −2.32550265534841404953396035599, 0.838397789147471146940324367828, 1.98999239426155435609453632854, 3.91618841094296413469633418879, 4.67254464989689035797190970781, 5.26086048999951567280937729705, 6.21149392684217801921701370378, 7.31712898765692184362250342882, 8.348655850654230741267742183934, 9.080558042685021410989660893620, 9.690559355578118747714529544166

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