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Properties

Label 1-832-832.413-r0-0-0
Degree $1$
Conductor $832$
Sign $0.239 + 0.970i$
Analytic cond. $3.86379$
Root an. cond. $3.86379$
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  + (−0.793 + 0.608i)3-s + (0.382 + 0.923i)5-s + (0.965 + 0.258i)7-s + (0.258 − 0.965i)9-s + (−0.130 + 0.991i)11-s + (−0.866 − 0.5i)15-s + (0.866 − 0.5i)17-s + (0.991 − 0.130i)19-s + (−0.923 + 0.382i)21-s + (0.965 − 0.258i)23-s + (−0.707 + 0.707i)25-s + (0.382 + 0.923i)27-s + (0.793 − 0.608i)29-s + 31-s + (−0.5 − 0.866i)33-s + ⋯
L(s)  = 1  + (−0.793 + 0.608i)3-s + (0.382 + 0.923i)5-s + (0.965 + 0.258i)7-s + (0.258 − 0.965i)9-s + (−0.130 + 0.991i)11-s + (−0.866 − 0.5i)15-s + (0.866 − 0.5i)17-s + (0.991 − 0.130i)19-s + (−0.923 + 0.382i)21-s + (0.965 − 0.258i)23-s + (−0.707 + 0.707i)25-s + (0.382 + 0.923i)27-s + (0.793 − 0.608i)29-s + 31-s + (−0.5 − 0.866i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.239 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.239 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(832\)    =    \(2^{6} \cdot 13\)
Sign: $0.239 + 0.970i$
Analytic conductor: \(3.86379\)
Root analytic conductor: \(3.86379\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{832} (413, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 832,\ (0:\ ),\ 0.239 + 0.970i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.170240437 + 0.9166009064i\)
\(L(\frac12)\) \(\approx\) \(1.170240437 + 0.9166009064i\)
\(L(1)\) \(\approx\) \(1.007783015 + 0.4151094217i\)
\(L(1)\) \(\approx\) \(1.007783015 + 0.4151094217i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 \)
good3 \( 1 + (-0.793 + 0.608i)T \)
5 \( 1 + (0.382 + 0.923i)T \)
7 \( 1 + (0.965 + 0.258i)T \)
11 \( 1 + (-0.130 + 0.991i)T \)
17 \( 1 + (0.866 - 0.5i)T \)
19 \( 1 + (0.991 - 0.130i)T \)
23 \( 1 + (0.965 - 0.258i)T \)
29 \( 1 + (0.793 - 0.608i)T \)
31 \( 1 + T \)
37 \( 1 + (0.991 + 0.130i)T \)
41 \( 1 + (-0.965 + 0.258i)T \)
43 \( 1 + (-0.793 - 0.608i)T \)
47 \( 1 - iT \)
53 \( 1 + (-0.923 + 0.382i)T \)
59 \( 1 + (0.608 - 0.793i)T \)
61 \( 1 + (-0.130 - 0.991i)T \)
67 \( 1 + (0.793 - 0.608i)T \)
71 \( 1 + (0.965 + 0.258i)T \)
73 \( 1 + (-0.707 - 0.707i)T \)
79 \( 1 - iT \)
83 \( 1 + (-0.382 + 0.923i)T \)
89 \( 1 + (0.258 + 0.965i)T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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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

−21.78478956096077774298867893299, −21.32994579689292108201199711387, −20.50213460952264812463414282313, −19.51490483973111138511956704229, −18.67695144404241724934201521437, −17.85964277525553615710809760176, −17.19296270962182246649822024460, −16.5721835468280404126886231438, −15.87004349967626453869943764370, −14.4959174032034389015612356691, −13.70671384186247945763768449442, −13.07651200150108210506590740917, −12.110175539832623564791900219116, −11.4962394086758655125585311042, −10.648425059750945792139140125335, −9.702371181441959774541531576844, −8.396920781086312385774406544225, −7.99855176938073915835801347801, −6.866948518496860382106786114708, −5.73978241194643010016354396631, −5.25436337934858772438988853877, −4.38670709239280119153536010860, −2.900342857647643669587095701863, −1.38116762238238609877955350013, −1.03350912204553819271675135023, 1.17862425971535528610269549999, 2.46125614666238779839206906956, 3.48914537901513768167891924687, 4.81018992223219793112417372878, 5.211958805820074368581505732842, 6.364614313019398152840945918275, 7.13892987728069699230197332429, 8.110100481984171534660551204982, 9.51453525222394704219250609233, 9.98953743515066638281275274351, 10.86773487887885666732272633818, 11.65102172382314225025506235147, 12.20029159299224876960781824152, 13.54216976783287194339733599350, 14.460999832946764126105390637120, 15.10937919676788576189666675318, 15.72506670896459947929405585043, 16.95006838769310898287833489735, 17.50570310757847668107198924684, 18.25877995517541821588652076636, 18.74737147528649066424768870878, 20.23708050643598097112960575495, 20.903437371758170872855641982555, 21.59279305880165448494896504669, 22.28753518444136295809228561238

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