Properties

Label 2-1183-13.12-c1-0-59
Degree $2$
Conductor $1183$
Sign $0.554 + 0.832i$
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  + 2.22i·2-s + 0.549·3-s − 2.92·4-s − 4.22i·5-s + 1.22i·6-s + i·7-s − 2.06i·8-s − 2.69·9-s + 9.36·10-s + 0.549i·11-s − 1.60·12-s − 2.22·14-s − 2.31i·15-s − 1.28·16-s + 2.37·17-s − 5.98i·18-s + ⋯
L(s)  = 1  + 1.56i·2-s + 0.317·3-s − 1.46·4-s − 1.88i·5-s + 0.498i·6-s + 0.377i·7-s − 0.728i·8-s − 0.899·9-s + 2.96·10-s + 0.165i·11-s − 0.464·12-s − 0.593·14-s − 0.598i·15-s − 0.320·16-s + 0.576·17-s − 1.41i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.554 + 0.832i$
Analytic conductor: \(9.44630\)
Root analytic conductor: \(3.07348\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1183,\ (\ :1/2),\ 0.554 + 0.832i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6891263636\)
\(L(\frac12)\) \(\approx\) \(0.6891263636\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 - iT \)
13 \( 1 \)
good2 \( 1 - 2.22iT - 2T^{2} \)
3 \( 1 - 0.549T + 3T^{2} \)
5 \( 1 + 4.22iT - 5T^{2} \)
11 \( 1 - 0.549iT - 11T^{2} \)
17 \( 1 - 2.37T + 17T^{2} \)
19 \( 1 + 3.61iT - 19T^{2} \)
23 \( 1 + 5.81T + 23T^{2} \)
29 \( 1 + 3.59T + 29T^{2} \)
31 \( 1 + 5.14iT - 31T^{2} \)
37 \( 1 - 0.329iT - 37T^{2} \)
41 \( 1 + 6.29iT - 41T^{2} \)
43 \( 1 + 3.22T + 43T^{2} \)
47 \( 1 + 8.20iT - 47T^{2} \)
53 \( 1 - 2.65T + 53T^{2} \)
59 \( 1 - 1.80iT - 59T^{2} \)
61 \( 1 - 0.609T + 61T^{2} \)
67 \( 1 - 10.3iT - 67T^{2} \)
71 \( 1 + 11.1iT - 71T^{2} \)
73 \( 1 + 4.90iT - 73T^{2} \)
79 \( 1 + 14.0T + 79T^{2} \)
83 \( 1 + 5.73iT - 83T^{2} \)
89 \( 1 - 7.46iT - 89T^{2} \)
97 \( 1 + 6.84iT - 97T^{2} \)
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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

−9.097445927867216656783678059541, −8.676925204716452761292030533410, −8.077393745146981867233621790219, −7.36855432376593242514391273212, −6.02403088277351423273605615847, −5.54898181080703554846974630671, −4.85571104418603014938920748418, −3.90088120398658949254749553004, −2.10858386739300633156347135544, −0.26772572594525316824304681138, 1.77481783129762205720600333027, 2.80031010405854466933858165534, 3.33189292157050453225091944178, 4.06603378116059055414201340466, 5.68684983047049584526945624676, 6.55063514696501701817312474353, 7.56865041467491066494608536648, 8.358932539305314319983662478075, 9.595339848450707508653893009574, 10.09561198270610859367309771999

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