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Properties

Label 2-1183-7.4-c1-0-22
Degree $2$
Conductor $1183$
Sign $0.0722 + 0.997i$
Analytic cond. $9.44630$
Root an. cond. $3.07348$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.15 − 1.99i)2-s + (0.736 − 1.27i)3-s + (−1.65 + 2.86i)4-s + (−0.423 − 0.733i)5-s − 3.39·6-s + (−1.00 + 2.44i)7-s + 3.00·8-s + (0.414 + 0.718i)9-s + (−0.975 + 1.69i)10-s + (0.751 − 1.30i)11-s + (2.43 + 4.21i)12-s + (6.03 − 0.824i)14-s − 1.24·15-s + (−0.156 − 0.271i)16-s + (−1.03 + 1.79i)17-s + (0.954 − 1.65i)18-s + ⋯
L(s)  = 1  + (−0.814 − 1.41i)2-s + (0.425 − 0.736i)3-s + (−0.826 + 1.43i)4-s + (−0.189 − 0.328i)5-s − 1.38·6-s + (−0.378 + 0.925i)7-s + 1.06·8-s + (0.138 + 0.239i)9-s + (−0.308 + 0.534i)10-s + (0.226 − 0.392i)11-s + (0.702 + 1.21i)12-s + (1.61 − 0.220i)14-s − 0.322·15-s + (−0.0391 − 0.0678i)16-s + (−0.251 + 0.435i)17-s + (0.225 − 0.389i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0722 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0722 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.0722 + 0.997i$
Analytic conductor: \(9.44630\)
Root analytic conductor: \(3.07348\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (508, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1183,\ (\ :1/2),\ 0.0722 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.050610506\)
\(L(\frac12)\) \(\approx\) \(1.050610506\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (1.00 - 2.44i)T \)
13 \( 1 \)
good2 \( 1 + (1.15 + 1.99i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-0.736 + 1.27i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.423 + 0.733i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.751 + 1.30i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.03 - 1.79i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.0237 - 0.0410i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.90 - 6.77i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.35T + 29T^{2} \)
31 \( 1 + (3.93 - 6.80i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.35 + 5.80i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 10.0T + 41T^{2} \)
43 \( 1 - 9.26T + 43T^{2} \)
47 \( 1 + (0.180 + 0.311i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.35 - 2.34i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.820 - 1.42i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.26 + 3.91i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.02 + 1.76i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 14.2T + 71T^{2} \)
73 \( 1 + (3.38 - 5.85i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.82 + 10.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 11.5T + 83T^{2} \)
89 \( 1 + (-8.75 - 15.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 0.426T + 97T^{2} \)
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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

−9.269035835397579805584281536725, −9.064381786313452203630652329590, −8.239200231284383646849405046022, −7.48914821877688630134648675185, −6.35378259398214606935197119742, −5.20890217674358115595604068904, −3.83797849260478319698991665655, −2.87449420567668715766015219707, −2.06335327596758906375148509860, −1.03187737637536840615362094211, 0.73790263979146725920410097600, 2.94173369021530963250005992746, 4.07890672426305511940525885495, 4.84638264625884720513751936511, 6.13417147506604733333179961004, 6.90905751345672838610937347759, 7.35237221061009524751605328246, 8.306499823883562303774019427002, 9.246805240874392034048277911838, 9.520466242248474150164509118084

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