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Properties

Label 2-1183-7.2-c1-0-42
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: $\chi_{1183} (170, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1183,\ (\ :1/2),\ 0.946 + 0.321i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7129218781\)
\(L(\frac12)\) \(\approx\) \(0.7129218781\)
\(L(\frac{3}{2})\) 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

−9.557896233962768275746957640873, −8.466866005636207118483866406972, −7.74558484485730747105210426253, −7.19163984250026397953019210788, −6.77915621117939573552155712138, −5.98018724360060746379171371752, −4.76364834268687396113701577559, −3.34304062072891275900950187316, −2.35870622453074145758492646443, −0.46947647501993651453156595848, 1.05034315818298038287857736484, 2.20458139642227993322079513345, 4.02024703669884344244424008220, 4.44476896925271428769659529485, 5.31784805979867398381896915852, 6.14305856942856429796526388781, 7.75956279425114542685502132730, 8.496722422785398971259938807928, 9.057436867972456383790315936055, 9.991970443701958862158879237978

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