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Properties

Label 1-1191-1191.413-r1-0-0
Degree $1$
Conductor $1191$
Sign $0.880 - 0.474i$
Analytic cond. $127.990$
Root an. cond. $127.990$
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  + (0.959 − 0.281i)2-s + (0.841 − 0.540i)4-s + (0.959 − 0.281i)5-s + (−0.959 − 0.281i)7-s + (0.654 − 0.755i)8-s + (0.841 − 0.540i)10-s + (0.654 + 0.755i)11-s + (0.841 − 0.540i)13-s − 14-s + (0.415 − 0.909i)16-s + (0.959 + 0.281i)17-s + (0.415 + 0.909i)19-s + (0.654 − 0.755i)20-s + (0.841 + 0.540i)22-s + (0.654 + 0.755i)23-s + ⋯
L(s)  = 1  + (0.959 − 0.281i)2-s + (0.841 − 0.540i)4-s + (0.959 − 0.281i)5-s + (−0.959 − 0.281i)7-s + (0.654 − 0.755i)8-s + (0.841 − 0.540i)10-s + (0.654 + 0.755i)11-s + (0.841 − 0.540i)13-s − 14-s + (0.415 − 0.909i)16-s + (0.959 + 0.281i)17-s + (0.415 + 0.909i)19-s + (0.654 − 0.755i)20-s + (0.841 + 0.540i)22-s + (0.654 + 0.755i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1191 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.880 - 0.474i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1191 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.880 - 0.474i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1191\)    =    \(3 \cdot 397\)
Sign: $0.880 - 0.474i$
Analytic conductor: \(127.990\)
Root analytic conductor: \(127.990\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1191} (413, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1191,\ (1:\ ),\ 0.880 - 0.474i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(5.750872464 - 1.449712363i\)
\(L(\frac12)\) \(\approx\) \(5.750872464 - 1.449712363i\)
\(L(1)\) \(\approx\) \(2.410359564 - 0.5169758215i\)
\(L(1)\) \(\approx\) \(2.410359564 - 0.5169758215i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
397 \( 1 \)
good2 \( 1 + (0.959 - 0.281i)T \)
5 \( 1 + (0.959 - 0.281i)T \)
7 \( 1 + (-0.959 - 0.281i)T \)
11 \( 1 + (0.654 + 0.755i)T \)
13 \( 1 + (0.841 - 0.540i)T \)
17 \( 1 + (0.959 + 0.281i)T \)
19 \( 1 + (0.415 + 0.909i)T \)
23 \( 1 + (0.654 + 0.755i)T \)
29 \( 1 + (0.959 - 0.281i)T \)
31 \( 1 + (-0.142 + 0.989i)T \)
37 \( 1 + (-0.142 + 0.989i)T \)
41 \( 1 - T \)
43 \( 1 + (-0.959 + 0.281i)T \)
47 \( 1 + (-0.841 + 0.540i)T \)
53 \( 1 + (-0.415 + 0.909i)T \)
59 \( 1 + (0.142 - 0.989i)T \)
61 \( 1 + (0.841 + 0.540i)T \)
67 \( 1 + (-0.142 + 0.989i)T \)
71 \( 1 + (0.142 - 0.989i)T \)
73 \( 1 + (0.415 - 0.909i)T \)
79 \( 1 + T \)
83 \( 1 + (0.959 - 0.281i)T \)
89 \( 1 + (-0.841 - 0.540i)T \)
97 \( 1 + (-0.959 - 0.281i)T \)
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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

−21.32803438365538820654808471164, −20.591088913863660046122522334635, −19.589938719365169083415722833357, −18.82034284825899795067115858215, −17.98896212318604284772547744935, −16.76907016451204689249805766723, −16.544782137852408268764406227296, −15.63037829225957240406680785277, −14.70583315433689562167662551391, −13.952491466251770978292784433093, −13.44598074759306863975407633019, −12.72814636976668030539449343782, −11.763858179488749206600219896068, −11.05058275824413012171070776953, −10.06536277321304679392007273241, −9.164757205914330462106125945299, −8.36907627973858850522048542748, −6.85807659867857341282748621295, −6.603755174284260606400716980012, −5.748178373614252705203972896146, −5.02538967253784228206766237936, −3.68377852045093567421586309616, −3.11296887984661343496659368034, −2.18601455490251768899288491290, −0.939733531512079264389580561134, 1.11417731664985149450237867520, 1.62741371410601282216562386706, 3.07653080279380468680485855376, 3.51354500618173895336444901685, 4.73033853896166290014968937705, 5.55277518649245638163587978503, 6.326874793367499944719333443167, 6.897248417071358679951635535547, 8.13353703082883709431783018852, 9.436699347878894818534869834314, 10.01780748343518323854795282657, 10.58479271052784560885163868050, 11.8710444391021279449873748038, 12.47356232925551401164599865109, 13.21391764688135524032313509361, 13.79540747115919700226878712114, 14.55845418738480216919979344838, 15.433254421225460811117051346439, 16.30365046225365098093095295792, 16.89986960369035588786035679780, 17.84732997008762171370889248326, 18.85626838404680524461937250387, 19.623295801236289023226433698707, 20.43938333608944099451543507816, 20.91337668551979434870338360693

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