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Properties

Label 2-1183-7.4-c1-0-72
Degree $2$
Conductor $1183$
Sign $-0.758 + 0.651i$
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  + (−0.593 − 1.02i)2-s + (1.46 − 2.54i)3-s + (0.294 − 0.510i)4-s + (1.70 + 2.94i)5-s − 3.48·6-s + (2.56 − 0.653i)7-s − 3.07·8-s + (−2.80 − 4.85i)9-s + (2.02 − 3.50i)10-s + (1.63 − 2.82i)11-s + (−0.865 − 1.49i)12-s + (−2.19 − 2.24i)14-s + 9.98·15-s + (1.23 + 2.14i)16-s + (1.27 − 2.21i)17-s + (−3.33 + 5.77i)18-s + ⋯
L(s)  = 1  + (−0.419 − 0.727i)2-s + (0.847 − 1.46i)3-s + (0.147 − 0.255i)4-s + (0.761 + 1.31i)5-s − 1.42·6-s + (0.969 − 0.246i)7-s − 1.08·8-s + (−0.935 − 1.61i)9-s + (0.639 − 1.10i)10-s + (0.492 − 0.853i)11-s + (−0.249 − 0.432i)12-s + (−0.586 − 0.601i)14-s + 2.57·15-s + (0.309 + 0.535i)16-s + (0.310 − 0.537i)17-s + (−0.785 + 1.36i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.758 + 0.651i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.758 + 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.758 + 0.651i$
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: \((2,\ 1183,\ (\ :1/2),\ -0.758 + 0.651i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.326150230\)
\(L(\frac12)\) \(\approx\) \(2.326150230\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.56 + 0.653i)T \)
13 \( 1 \)
good2 \( 1 + (0.593 + 1.02i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-1.46 + 2.54i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.70 - 2.94i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.63 + 2.82i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.27 + 2.21i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.48 - 4.30i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.938 - 1.62i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 0.273T + 29T^{2} \)
31 \( 1 + (0.341 - 0.591i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.60 + 9.71i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 1.85T + 41T^{2} \)
43 \( 1 + 0.826T + 43T^{2} \)
47 \( 1 + (-4.75 - 8.23i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.20 - 3.82i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.96 - 3.40i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.60 + 6.24i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.42 - 5.92i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12.1T + 71T^{2} \)
73 \( 1 + (1.43 - 2.48i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (8.13 + 14.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6.44T + 83T^{2} \)
89 \( 1 + (-5.09 - 8.82i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 1.21T + 97T^{2} \)
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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

−9.388763236257602249878670036315, −8.739806409404733485442113630959, −7.68768602057078644272196132621, −7.15169011575156960464220091620, −6.22147600795339819300807196483, −5.64183050338466523440934085837, −3.47931866400448935785683495172, −2.76419107141657254058234078639, −1.87379228095170794693690702124, −1.15614330624728775652193573990, 1.75027935476524131167508291096, 2.98988002287487972062252408844, 4.25783260074240255234046095719, 4.92578966187062625101218821836, 5.60660713844322915601477782268, 6.94168427322540081669904680223, 8.056121366874973536126048104910, 8.608119657755677320070049194645, 9.084671931989798855294436004380, 9.675568632271488940389997849198

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