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Properties

Label 2-1183-7.4-c1-0-60
Degree $2$
Conductor $1183$
Sign $0.782 + 0.622i$
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  + (−0.0978 − 0.169i)2-s + (0.129 − 0.224i)3-s + (0.980 − 1.69i)4-s + (1.96 + 3.40i)5-s − 0.0508·6-s + (−1.12 − 2.39i)7-s − 0.775·8-s + (1.46 + 2.53i)9-s + (0.384 − 0.666i)10-s + (2.25 − 3.90i)11-s + (−0.254 − 0.441i)12-s + (−0.296 + 0.424i)14-s + 1.02·15-s + (−1.88 − 3.26i)16-s + (1.14 − 1.97i)17-s + (0.286 − 0.496i)18-s + ⋯
L(s)  = 1  + (−0.0691 − 0.119i)2-s + (0.0749 − 0.129i)3-s + (0.490 − 0.849i)4-s + (0.879 + 1.52i)5-s − 0.0207·6-s + (−0.424 − 0.905i)7-s − 0.274·8-s + (0.488 + 0.846i)9-s + (0.121 − 0.210i)10-s + (0.679 − 1.17i)11-s + (−0.0735 − 0.127i)12-s + (−0.0791 + 0.113i)14-s + 0.263·15-s + (−0.471 − 0.816i)16-s + (0.276 − 0.479i)17-s + (0.0676 − 0.117i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.782 + 0.622i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.782 + 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.782 + 0.622i$
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: \((2,\ 1183,\ (\ :1/2),\ 0.782 + 0.622i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.172412396\)
\(L(\frac12)\) \(\approx\) \(2.172412396\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (1.12 + 2.39i)T \)
13 \( 1 \)
good2 \( 1 + (0.0978 + 0.169i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-0.129 + 0.224i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.96 - 3.40i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.25 + 3.90i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.14 + 1.97i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.893 + 1.54i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.870 + 1.50i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.65T + 29T^{2} \)
31 \( 1 + (-2.80 + 4.85i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.57 - 6.18i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 8.11T + 41T^{2} \)
43 \( 1 - 6.81T + 43T^{2} \)
47 \( 1 + (-1.77 - 3.07i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.64 - 2.84i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.25 + 3.90i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.77 + 6.53i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.33 - 10.9i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 9.54T + 71T^{2} \)
73 \( 1 + (-0.540 + 0.935i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.395 + 0.685i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 7.14T + 83T^{2} \)
89 \( 1 + (5.63 + 9.75i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 8.81T + 97T^{2} \)
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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

−9.951163953997881757171244791749, −9.205544515227742055776936749470, −7.73213253972311882436284243642, −7.04902581474394039427485798833, −6.28889255483472431385778673848, −5.88783115703863361592759957931, −4.46112567481647872614484643317, −3.10648867390628805538442270541, −2.39366174369050923807680320537, −1.08085235607704099021025455737, 1.41936383797075055609014222901, 2.42198830365041545671222303839, 3.82690475100046185980516466086, 4.59965980353679761288432900475, 5.81174111019487325112350830362, 6.38093287266371514440764044623, 7.41693583830459589230879616511, 8.431345546218477500453379797646, 9.199651537263454694053254893027, 9.418218984995217491355471790085

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