Properties

Label 2-1183-13.12-c1-0-22
Degree $2$
Conductor $1183$
Sign $0.0304 + 0.999i$
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  − 1.93i·2-s − 2.54·3-s − 1.74·4-s − 0.312i·5-s + 4.92i·6-s i·7-s − 0.495i·8-s + 3.48·9-s − 0.603·10-s + 4.16i·11-s + 4.43·12-s − 1.93·14-s + 0.794i·15-s − 4.44·16-s + 5.20·17-s − 6.73i·18-s + ⋯
L(s)  = 1  − 1.36i·2-s − 1.46·3-s − 0.871·4-s − 0.139i·5-s + 2.01i·6-s − 0.377i·7-s − 0.175i·8-s + 1.16·9-s − 0.190·10-s + 1.25i·11-s + 1.28·12-s − 0.517·14-s + 0.205i·15-s − 1.11·16-s + 1.26·17-s − 1.58i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0304 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0304 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.0304 + 0.999i$
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: \((2,\ 1183,\ (\ :1/2),\ 0.0304 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9257927855\)
\(L(\frac12)\) \(\approx\) \(0.9257927855\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.93iT - 2T^{2} \)
3 \( 1 + 2.54T + 3T^{2} \)
5 \( 1 + 0.312iT - 5T^{2} \)
11 \( 1 - 4.16iT - 11T^{2} \)
17 \( 1 - 5.20T + 17T^{2} \)
19 \( 1 - 4.87iT - 19T^{2} \)
23 \( 1 + 3.39T + 23T^{2} \)
29 \( 1 - 3.54T + 29T^{2} \)
31 \( 1 - 9.52iT - 31T^{2} \)
37 \( 1 + 11.7iT - 37T^{2} \)
41 \( 1 - 0.433iT - 41T^{2} \)
43 \( 1 - 8.96T + 43T^{2} \)
47 \( 1 - 8.62iT - 47T^{2} \)
53 \( 1 + 1.14T + 53T^{2} \)
59 \( 1 + 11.7iT - 59T^{2} \)
61 \( 1 + 4.87T + 61T^{2} \)
67 \( 1 + 8.71iT - 67T^{2} \)
71 \( 1 + 6.09iT - 71T^{2} \)
73 \( 1 + 3.59iT - 73T^{2} \)
79 \( 1 - 9.90T + 79T^{2} \)
83 \( 1 - 0.500iT - 83T^{2} \)
89 \( 1 - 10.5iT - 89T^{2} \)
97 \( 1 - 2.35iT - 97T^{2} \)
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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

−9.969531702849790185144800176701, −9.279363190392574526272463153878, −7.83564872669325909816281979445, −6.96730793526039337244710727710, −6.08710846472854142314662871060, −5.06395344290899323628732408148, −4.34539417571230829540726425970, −3.31686402516441776766854912673, −1.83921944994878094277127535548, −0.860582900003912876305567097968, 0.72632906423735141949661504142, 2.83903052340138230685223863982, 4.43042592150593150938547526846, 5.33657863216841373739710636188, 5.86360740805907588753224644611, 6.41576880029613254467008494159, 7.23139907203513451870325329842, 8.152738865665400423062667469946, 8.866935750636166803855512619921, 9.997439478968086186045780671362

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