Properties

Label 2-1183-91.62-c0-0-5
Degree $2$
Conductor $1183$
Sign $-0.0502 + 0.998i$
Analytic cond. $0.590393$
Root an. cond. $0.768370$
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  + (1.56 − 0.900i)2-s + (1.12 − 1.94i)4-s + (−0.866 − 0.5i)7-s − 2.24i·8-s + (−0.5 + 0.866i)9-s + (1.07 − 0.623i)11-s − 1.80·14-s + (−0.900 − 1.56i)16-s + 1.80i·18-s + (1.12 − 1.94i)22-s + (−0.222 − 0.385i)23-s − 25-s + (−1.94 + 1.12i)28-s + (0.900 + 1.56i)29-s + (−0.866 − 0.500i)32-s + ⋯
L(s)  = 1  + (1.56 − 0.900i)2-s + (1.12 − 1.94i)4-s + (−0.866 − 0.5i)7-s − 2.24i·8-s + (−0.5 + 0.866i)9-s + (1.07 − 0.623i)11-s − 1.80·14-s + (−0.900 − 1.56i)16-s + 1.80i·18-s + (1.12 − 1.94i)22-s + (−0.222 − 0.385i)23-s − 25-s + (−1.94 + 1.12i)28-s + (0.900 + 1.56i)29-s + (−0.866 − 0.500i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0502 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0502 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.0502 + 0.998i$
Analytic conductor: \(0.590393\)
Root analytic conductor: \(0.768370\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (699, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1183,\ (\ :0),\ -0.0502 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.259233437\)
\(L(\frac12)\) \(\approx\) \(2.259233437\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 \)
good2 \( 1 + (-1.56 + 0.900i)T + (0.5 - 0.866i)T^{2} \)
3 \( 1 + (0.5 - 0.866i)T^{2} \)
5 \( 1 + T^{2} \)
11 \( 1 + (-1.07 + 0.623i)T + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.222 + 0.385i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.900 - 1.56i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (0.385 - 0.222i)T + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.900 - 1.56i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + 0.445T + T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.385 + 0.222i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.385 + 0.222i)T + (0.5 + 0.866i)T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - 1.24T + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T^{2} \)
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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

−10.14712218980951444014893653361, −9.214283085125895941472573255075, −8.091914249678973365617626000867, −6.75570992410048535535773027496, −6.23088715286458713420184627815, −5.30266348973670830105220811097, −4.40351161863083473325251855968, −3.51609540835881232749599847832, −2.83340053426143836574248479789, −1.50361970826528540827351689115, 2.37704253381719336701684406717, 3.54053353275317019788777705355, 4.02039784765041429902795604060, 5.20646043187583632361579894760, 6.12718972223620377390796361162, 6.46056485089736677950921781566, 7.29561611769595736008861179962, 8.344905381722920352869146329962, 9.268386728702999664991733709872, 10.00912549386231903036586926334

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