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Properties

Label 1-1183-1183.1096-r0-0-0
Degree $1$
Conductor $1183$
Sign $0.999 + 0.0388i$
Analytic cond. $5.49382$
Root an. cond. $5.49382$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.748 − 0.663i)2-s + (0.987 + 0.160i)3-s + (0.120 − 0.992i)4-s + (−0.278 + 0.960i)5-s + (0.845 − 0.534i)6-s + (−0.568 − 0.822i)8-s + (0.948 + 0.316i)9-s + (0.428 + 0.903i)10-s + (−0.948 + 0.316i)11-s + (0.278 − 0.960i)12-s + (−0.428 + 0.903i)15-s + (−0.970 − 0.239i)16-s + (0.568 + 0.822i)17-s + (0.919 − 0.391i)18-s + (0.5 + 0.866i)19-s + (0.919 + 0.391i)20-s + ⋯
L(s)  = 1  + (0.748 − 0.663i)2-s + (0.987 + 0.160i)3-s + (0.120 − 0.992i)4-s + (−0.278 + 0.960i)5-s + (0.845 − 0.534i)6-s + (−0.568 − 0.822i)8-s + (0.948 + 0.316i)9-s + (0.428 + 0.903i)10-s + (−0.948 + 0.316i)11-s + (0.278 − 0.960i)12-s + (−0.428 + 0.903i)15-s + (−0.970 − 0.239i)16-s + (0.568 + 0.822i)17-s + (0.919 − 0.391i)18-s + (0.5 + 0.866i)19-s + (0.919 + 0.391i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0388i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0388i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.999 + 0.0388i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (1096, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ 0.999 + 0.0388i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.087683086 + 0.06001046875i\)
\(L(\frac12)\) \(\approx\) \(3.087683086 + 0.06001046875i\)
\(L(1)\) \(\approx\) \(2.010706360 - 0.2416865069i\)
\(L(1)\) \(\approx\) \(2.010706360 - 0.2416865069i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.748 - 0.663i)T \)
3 \( 1 + (0.987 + 0.160i)T \)
5 \( 1 + (-0.278 + 0.960i)T \)
11 \( 1 + (-0.948 + 0.316i)T \)
17 \( 1 + (0.568 + 0.822i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + T \)
29 \( 1 + (0.948 + 0.316i)T \)
31 \( 1 + (0.0402 + 0.999i)T \)
37 \( 1 + (-0.885 - 0.464i)T \)
41 \( 1 + (0.632 - 0.774i)T \)
43 \( 1 + (-0.0402 + 0.999i)T \)
47 \( 1 + (0.919 + 0.391i)T \)
53 \( 1 + (0.428 - 0.903i)T \)
59 \( 1 + (0.970 - 0.239i)T \)
61 \( 1 + (-0.996 - 0.0804i)T \)
67 \( 1 + (0.919 + 0.391i)T \)
71 \( 1 + (-0.987 - 0.160i)T \)
73 \( 1 + (0.200 - 0.979i)T \)
79 \( 1 + (-0.919 - 0.391i)T \)
83 \( 1 + (0.354 - 0.935i)T \)
89 \( 1 - T \)
97 \( 1 + (-0.692 + 0.721i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.05828494214715175977118132110, −20.6681849069477366962257440631, −19.94033212497442033223985238550, −18.95725562032920900915445138155, −18.137688602314443764677930843030, −17.13907727094741799081719628988, −16.339559675108215737076784474723, −15.53243833792589505180031118654, −15.27941640888014134050176752033, −13.98026519105416374320828752220, −13.58960926875377342974510308489, −12.87962880310677456534839439093, −12.17926122869567791095808997274, −11.28646343778226599993376338671, −9.89712696634670495418033726199, −8.952600220024400500456812881563, −8.3907307641065857144175347997, −7.569361405281308745979639113324, −7.01609283625633804319218427809, −5.65232234203827930108658342838, −4.92842431148641446921672538162, −4.16134623849574388775524867302, −3.09599317206332470947305507268, −2.47785226457273583932405356489, −0.90715001023739537060519171515, 1.39122096361615732383157375224, 2.40100641618198718878276027534, 3.13154420894973236616131782246, 3.71440179312039918539160945229, 4.73136122979808278627002712385, 5.69235769694340511556081476761, 6.83552974168217851091064462256, 7.55821850742405288728579027181, 8.51886453551192846113692493483, 9.62947261006923146673199314001, 10.4784141306917761839943227649, 10.67474157856270042145851175920, 12.01734117581139692723596163412, 12.6849016559574389974253961637, 13.511703680271634295531964470120, 14.40695532658922664725218168232, 14.65135878395680275995369129850, 15.61205061760933303090310666547, 16.06462497942279204401909898686, 17.73592135882541461044488439661, 18.53034739482859979014216728282, 19.168859224659491842450454821896, 19.63665965156374982760335787024, 20.64919080006247529752501806348, 21.16926200882092610133293185946

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