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Properties

Label 1-63-63.59-r0-0-0
Degree $1$
Conductor $63$
Sign $0.220 + 0.975i$
Analytic cond. $0.292570$
Root an. cond. $0.292570$
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  + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)4-s + 5-s − 8-s + (0.5 + 0.866i)10-s − 11-s + (0.5 + 0.866i)13-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + (0.5 − 0.866i)19-s + (−0.5 + 0.866i)20-s + (−0.5 − 0.866i)22-s − 23-s + 25-s + (−0.5 + 0.866i)26-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)4-s + 5-s − 8-s + (0.5 + 0.866i)10-s − 11-s + (0.5 + 0.866i)13-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + (0.5 − 0.866i)19-s + (−0.5 + 0.866i)20-s + (−0.5 − 0.866i)22-s − 23-s + 25-s + (−0.5 + 0.866i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.220 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.220 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.220 + 0.975i$
Analytic conductor: \(0.292570\)
Root analytic conductor: \(0.292570\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 63,\ (0:\ ),\ 0.220 + 0.975i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9246000774 + 0.7389214309i\)
\(L(\frac12)\) \(\approx\) \(0.9246000774 + 0.7389214309i\)
\(L(1)\) \(\approx\) \(1.128486902 + 0.6128756861i\)
\(L(1)\) \(\approx\) \(1.128486902 + 0.6128756861i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + T \)
11 \( 1 - T \)
13 \( 1 + (0.5 + 0.866i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 - T \)
29 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (0.5 + 0.866i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (0.5 + 0.866i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 - T \)
73 \( 1 + (0.5 + 0.866i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
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

−32.09193275267005468644258924668, −30.87581044530678260037304082771, −29.89452479195626043137254302356, −28.918290603004894101166968210796, −28.1511854616944443825951575887, −26.69607153123995228985365304334, −25.40313895233850314491725025781, −24.119976462106064041466506408486, −22.94387545075650877601734934360, −21.82667676128428896271265481538, −20.949538978434060808929310964904, −19.965803152785516704850279079900, −18.4275588202031893173103732351, −17.70647902014491223000322470886, −15.821550469840632262863615144357, −14.39429907478334975277905858326, −13.3435816782873526370870201358, −12.443128203893929926387952131978, −10.70440998346537719772640162460, −10.02089714465557712392417062663, −8.43733659075593957149258872107, −6.15142857931156146692046655701, −5.10780251555827769058010396516, −3.25917734839547619543310972180, −1.76678857862172504074862294893, 2.61872943731627077232150342832, 4.59961756756476740947147722682, 5.837037919022833816700818316422, 7.03412509587865857482382169180, 8.589599239774548401858039205048, 9.859054166982454675418286960811, 11.7099699075301341968691935946, 13.3914758164834907488565494856, 13.80173475139901737097695762573, 15.384523882236952165887714910986, 16.40924876631651094765703779901, 17.67068723770706592885857782674, 18.49099288646183393028594722565, 20.6161731910180914356965966638, 21.53218885754053369282604119773, 22.553484940129894147375147565065, 23.840771946238037910134052511915, 24.72623205486849160390363114309, 25.959169955503958760113871421550, 26.48953971292605881443043315377, 28.25601120626909799413808653395, 29.38887840925855409252677547234, 30.64298297139703650960351127129, 31.65344711933989909607530137300, 32.759202567839123052866875717390

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