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Properties

Label 1-1183-1183.235-r0-0-0
Degree $1$
Conductor $1183$
Sign $0.908 - 0.418i$
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  + (0.278 − 0.960i)2-s + (0.799 − 0.600i)3-s + (−0.845 − 0.534i)4-s + (−0.996 + 0.0804i)5-s + (−0.354 − 0.935i)6-s + (−0.748 + 0.663i)8-s + (0.278 − 0.960i)9-s + (−0.200 + 0.979i)10-s + (0.278 + 0.960i)11-s + (−0.996 + 0.0804i)12-s + (−0.748 + 0.663i)15-s + (0.428 + 0.903i)16-s + (0.948 + 0.316i)17-s + (−0.845 − 0.534i)18-s + (−0.5 + 0.866i)19-s + (0.885 + 0.464i)20-s + ⋯
L(s)  = 1  + (0.278 − 0.960i)2-s + (0.799 − 0.600i)3-s + (−0.845 − 0.534i)4-s + (−0.996 + 0.0804i)5-s + (−0.354 − 0.935i)6-s + (−0.748 + 0.663i)8-s + (0.278 − 0.960i)9-s + (−0.200 + 0.979i)10-s + (0.278 + 0.960i)11-s + (−0.996 + 0.0804i)12-s + (−0.748 + 0.663i)15-s + (0.428 + 0.903i)16-s + (0.948 + 0.316i)17-s + (−0.845 − 0.534i)18-s + (−0.5 + 0.866i)19-s + (0.885 + 0.464i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.908 - 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.908 - 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.908 - 0.418i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ 0.908 - 0.418i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.370868353 - 0.3008455560i\)
\(L(\frac12)\) \(\approx\) \(1.370868353 - 0.3008455560i\)
\(L(1)\) \(\approx\) \(1.018401783 - 0.5406896121i\)
\(L(1)\) \(\approx\) \(1.018401783 - 0.5406896121i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.278 - 0.960i)T \)
3 \( 1 + (0.799 - 0.600i)T \)
5 \( 1 + (-0.996 + 0.0804i)T \)
11 \( 1 + (0.278 + 0.960i)T \)
17 \( 1 + (0.948 + 0.316i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.970 + 0.239i)T \)
31 \( 1 + (-0.632 + 0.774i)T \)
37 \( 1 + (0.987 + 0.160i)T \)
41 \( 1 + (0.120 + 0.992i)T \)
43 \( 1 + (-0.354 + 0.935i)T \)
47 \( 1 + (-0.845 + 0.534i)T \)
53 \( 1 + (0.948 + 0.316i)T \)
59 \( 1 + (0.428 - 0.903i)T \)
61 \( 1 + (0.948 - 0.316i)T \)
67 \( 1 + (-0.0402 - 0.999i)T \)
71 \( 1 + (0.120 + 0.992i)T \)
73 \( 1 + (0.278 + 0.960i)T \)
79 \( 1 + (-0.845 + 0.534i)T \)
83 \( 1 + (0.120 - 0.992i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (0.568 - 0.822i)T \)
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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

−21.34335078526612165000741239367, −20.603143249382775733385806702333, −19.688436536831298201614013521487, −18.935175145930051869561338995194, −18.3768971215023804962975243465, −16.93929410735849619189334491803, −16.42989943124156068064361841071, −15.90038652904939981673685458188, −14.883391904272380484376407350679, −14.70682623743246860509395814457, −13.65134410260608096860437327497, −13.01879397901631844514690063720, −11.957590576927510199348194058468, −11.08223896175154950272135073928, −10.00823839622689766632772805227, −8.975268784459596394343186585528, −8.52741436494394882038381138495, −7.7209599135183936706870876677, −7.06460238489577700359284900043, −5.84464834648366151532031730205, −4.96254296743546628457847000506, −3.98854963329051419077172786680, −3.59538889473242772037202740386, −2.523483029069629237887107880496, −0.5148239091230471171804107088, 1.21132454058538010244866718312, 1.93648346814705995473111793175, 3.11273253588395630867902308142, 3.733194875749672026047324341339, 4.45997685711003871623230412248, 5.70738096845547835767461258624, 6.85162763779216048649886157758, 7.828490382168661106460448221030, 8.336369549165910441644952384923, 9.47155167578442647071589656916, 9.994978249134426556904002368710, 11.19803973074580151574563566826, 11.89490936022244292351805214130, 12.64854826273666516894506019553, 13.011492116682925211462082440585, 14.3866345601713363875413494614, 14.59704165519618451151313730404, 15.37288608251647169736687047184, 16.56413713178155501014524318495, 17.708135212088285500790434320831, 18.46423537617390088357731868661, 19.0112245317753617023997304181, 19.876514316215713556698635008093, 20.08124222596602748284240776433, 20.99611418010381275292982306703

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