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Properties

Label 1-6369-6369.113-r0-0-0
Degree $1$
Conductor $6369$
Sign $0.647 - 0.761i$
Analytic cond. $29.5775$
Root an. cond. $29.5775$
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  + (−0.928 − 0.370i)2-s + (0.725 + 0.688i)4-s + (0.856 − 0.516i)5-s + (0.358 − 0.933i)7-s + (−0.418 − 0.908i)8-s + (−0.986 + 0.162i)10-s + (0.842 − 0.539i)13-s + (−0.678 + 0.734i)14-s + (0.0523 + 0.998i)16-s + (−0.424 + 0.905i)17-s + (−0.654 + 0.756i)19-s + (0.976 + 0.214i)20-s + (0.831 + 0.555i)23-s + (0.465 − 0.884i)25-s + (−0.982 + 0.188i)26-s + ⋯
L(s)  = 1  + (−0.928 − 0.370i)2-s + (0.725 + 0.688i)4-s + (0.856 − 0.516i)5-s + (0.358 − 0.933i)7-s + (−0.418 − 0.908i)8-s + (−0.986 + 0.162i)10-s + (0.842 − 0.539i)13-s + (−0.678 + 0.734i)14-s + (0.0523 + 0.998i)16-s + (−0.424 + 0.905i)17-s + (−0.654 + 0.756i)19-s + (0.976 + 0.214i)20-s + (0.831 + 0.555i)23-s + (0.465 − 0.884i)25-s + (−0.982 + 0.188i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6369 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.647 - 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6369 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.647 - 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6369\)    =    \(3 \cdot 11 \cdot 193\)
Sign: $0.647 - 0.761i$
Analytic conductor: \(29.5775\)
Root analytic conductor: \(29.5775\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6369} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6369,\ (0:\ ),\ 0.647 - 0.761i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.401040297 - 0.6477841914i\)
\(L(\frac12)\) \(\approx\) \(1.401040297 - 0.6477841914i\)
\(L(1)\) \(\approx\) \(0.8814979389 - 0.2564787723i\)
\(L(1)\) \(\approx\) \(0.8814979389 - 0.2564787723i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
193 \( 1 \)
good2 \( 1 + (-0.928 - 0.370i)T \)
5 \( 1 + (0.856 - 0.516i)T \)
7 \( 1 + (0.358 - 0.933i)T \)
13 \( 1 + (0.842 - 0.539i)T \)
17 \( 1 + (-0.424 + 0.905i)T \)
19 \( 1 + (-0.654 + 0.756i)T \)
23 \( 1 + (0.831 + 0.555i)T \)
29 \( 1 + (-0.664 + 0.747i)T \)
31 \( 1 + (-0.370 + 0.928i)T \)
37 \( 1 + (-0.494 - 0.869i)T \)
41 \( 1 + (0.613 + 0.789i)T \)
43 \( 1 + (0.707 + 0.707i)T \)
47 \( 1 + (0.835 + 0.550i)T \)
53 \( 1 + (-0.0457 - 0.998i)T \)
59 \( 1 + (-0.284 - 0.958i)T \)
61 \( 1 + (0.0849 + 0.996i)T \)
67 \( 1 + (0.555 - 0.831i)T \)
71 \( 1 + (-0.999 + 0.0196i)T \)
73 \( 1 + (0.979 + 0.201i)T \)
79 \( 1 + (-0.970 + 0.239i)T \)
83 \( 1 + (0.0130 - 0.999i)T \)
89 \( 1 + (0.773 - 0.634i)T \)
97 \( 1 + (-0.969 - 0.246i)T \)
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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

−17.645413090711619736355608956207, −17.21395469049359663459268509741, −16.559383592274004964050420792921, −15.61690141351611537960714608273, −15.29592179418460730784771400477, −14.624897306269825163289146286314, −13.872140138206451988615375765234, −13.33756892076780355101336738609, −12.330601957863765788281580626018, −11.451689298319583286442057932118, −11.05987872266497544325620609865, −10.460539027205245943767542711834, −9.50833153824449926983508584517, −9.02161481972697088388221787905, −8.713024168145675548403164513208, −7.675773955057146699049130326283, −6.94592037375288763563551148003, −6.39884908077319302540382801067, −5.72853879236264879271541921668, −5.17422930252879303719307834832, −4.171302306177090136280040975020, −2.83399694215775759089509647197, −2.36818147223813704368317770377, −1.77107882629026146751253248783, −0.728838900232409854819603891003, 0.71449207117977903148133344101, 1.52130673262916968211323890795, 1.83915383581031473949636335296, 3.03050702316492832075759867419, 3.74285729502472975875432892342, 4.46920613060345284077896877027, 5.52719058014134372352524609706, 6.168438232483389926142858470072, 6.91964394290501999284955162189, 7.66797491512865319267405697532, 8.38846268403353687591995426828, 8.87392331199932278244207173066, 9.58634999805423895681698504184, 10.35702383171997209424594127079, 10.83110937301555368296193784570, 11.20434888517156615281638821071, 12.46562581734599410245065478906, 12.839115196798386852927660878944, 13.36448489805453716846021367438, 14.28558125823557199065072712117, 14.89807835943478461395975711103, 15.92765689858812697829130567773, 16.37281477817792948377456999629, 17.10544293052377060250974822036, 17.52301760840144800522781339840

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