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Properties

Label 2-1183-7.4-c1-0-17
Degree $2$
Conductor $1183$
Sign $-0.946 + 0.321i$
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  + (0.672 + 1.16i)2-s + (−1.02 + 1.77i)3-s + (0.0951 − 0.164i)4-s + (1.78 + 3.08i)5-s − 2.75·6-s + (−2.62 − 0.349i)7-s + 2.94·8-s + (−0.601 − 1.04i)9-s + (−2.39 + 4.15i)10-s + (0.639 − 1.10i)11-s + (0.195 + 0.337i)12-s + (−1.35 − 3.29i)14-s − 7.31·15-s + (1.79 + 3.10i)16-s + (−3.86 + 6.70i)17-s + (0.809 − 1.40i)18-s + ⋯
L(s)  = 1  + (0.475 + 0.823i)2-s + (−0.591 + 1.02i)3-s + (0.0475 − 0.0824i)4-s + (0.797 + 1.38i)5-s − 1.12·6-s + (−0.991 − 0.132i)7-s + 1.04·8-s + (−0.200 − 0.347i)9-s + (−0.758 + 1.31i)10-s + (0.192 − 0.333i)11-s + (0.0563 + 0.0975i)12-s + (−0.362 − 0.879i)14-s − 1.88·15-s + (0.447 + 0.775i)16-s + (−0.938 + 1.62i)17-s + (0.190 − 0.330i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.946 + 0.321i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.946 + 0.321i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.946 + 0.321i$
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: \((2,\ 1183,\ (\ :1/2),\ -0.946 + 0.321i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.722183947\)
\(L(\frac12)\) \(\approx\) \(1.722183947\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.62 + 0.349i)T \)
13 \( 1 \)
good2 \( 1 + (-0.672 - 1.16i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (1.02 - 1.77i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.78 - 3.08i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.639 + 1.10i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (3.86 - 6.70i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.471 + 0.817i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.823 + 1.42i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 4.04T + 29T^{2} \)
31 \( 1 + (2.57 - 4.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.528 + 0.914i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 4.19T + 41T^{2} \)
43 \( 1 - 3.83T + 43T^{2} \)
47 \( 1 + (0.447 + 0.774i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.0399 + 0.0692i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.59 + 9.68i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.81 - 6.60i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.16 + 5.47i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 11.4T + 71T^{2} \)
73 \( 1 + (-0.380 + 0.658i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.42 - 2.47i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 2.32T + 83T^{2} \)
89 \( 1 + (3.78 + 6.56i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 0.478T + 97T^{2} \)
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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

−10.43060719489575248500772965881, −9.749195825707289093737605545950, −8.641812035902834669088825254218, −7.28143259106811513722946231017, −6.45846545269245346189472701040, −6.21520969440841719584759126732, −5.37866651142251291271361160983, −4.28380656594185373853131679775, −3.43314869575543380481747106773, −2.08124743509207743761620333776, 0.65991394040141980488462827838, 1.77155550054348178035711450194, 2.67346133156918703107487945585, 4.07919073249406243501141882939, 4.99571714995376889877778669710, 5.86379890998877222284616808213, 6.77623895864432906725052124677, 7.43641083581997841671795251500, 8.654122248342521856356351253020, 9.469719549005491749134176996864

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