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Properties

Label 1-507-507.305-r0-0-0
Degree $1$
Conductor $507$
Sign $0.688 + 0.725i$
Analytic cond. $2.35449$
Root an. cond. $2.35449$
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  + (−0.960 + 0.278i)2-s + (0.845 − 0.534i)4-s + (0.822 + 0.568i)5-s + (0.316 + 0.948i)7-s + (−0.663 + 0.748i)8-s + (−0.948 − 0.316i)10-s + (0.721 − 0.692i)11-s + (−0.568 − 0.822i)14-s + (0.428 − 0.903i)16-s + (0.948 − 0.316i)17-s + (0.866 + 0.5i)19-s + (0.999 + 0.0402i)20-s + (−0.5 + 0.866i)22-s + (−0.5 − 0.866i)23-s + (0.354 + 0.935i)25-s + ⋯
L(s)  = 1  + (−0.960 + 0.278i)2-s + (0.845 − 0.534i)4-s + (0.822 + 0.568i)5-s + (0.316 + 0.948i)7-s + (−0.663 + 0.748i)8-s + (−0.948 − 0.316i)10-s + (0.721 − 0.692i)11-s + (−0.568 − 0.822i)14-s + (0.428 − 0.903i)16-s + (0.948 − 0.316i)17-s + (0.866 + 0.5i)19-s + (0.999 + 0.0402i)20-s + (−0.5 + 0.866i)22-s + (−0.5 − 0.866i)23-s + (0.354 + 0.935i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.688 + 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.688 + 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.688 + 0.725i$
Analytic conductor: \(2.35449\)
Root analytic conductor: \(2.35449\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 507,\ (0:\ ),\ 0.688 + 0.725i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.080447446 + 0.4642959937i\)
\(L(\frac12)\) \(\approx\) \(1.080447446 + 0.4642959937i\)
\(L(1)\) \(\approx\) \(0.8993388245 + 0.2333928716i\)
\(L(1)\) \(\approx\) \(0.8993388245 + 0.2333928716i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.960 + 0.278i)T \)
5 \( 1 + (0.822 + 0.568i)T \)
7 \( 1 + (0.316 + 0.948i)T \)
11 \( 1 + (0.721 - 0.692i)T \)
17 \( 1 + (0.948 - 0.316i)T \)
19 \( 1 + (0.866 + 0.5i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.278 - 0.960i)T \)
31 \( 1 + (0.935 + 0.354i)T \)
37 \( 1 + (0.160 + 0.987i)T \)
41 \( 1 + (-0.391 - 0.919i)T \)
43 \( 1 + (-0.987 - 0.160i)T \)
47 \( 1 + (-0.464 - 0.885i)T \)
53 \( 1 + (0.748 + 0.663i)T \)
59 \( 1 + (0.903 - 0.428i)T \)
61 \( 1 + (-0.200 - 0.979i)T \)
67 \( 1 + (-0.534 + 0.845i)T \)
71 \( 1 + (-0.600 + 0.799i)T \)
73 \( 1 + (-0.239 - 0.970i)T \)
79 \( 1 + (0.885 - 0.464i)T \)
83 \( 1 + (-0.992 + 0.120i)T \)
89 \( 1 + (-0.866 + 0.5i)T \)
97 \( 1 + (0.0804 + 0.996i)T \)
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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

−23.76080121547492915706450416187, −22.52454054100283261123116361754, −21.48524437653374058784519043459, −20.838544150950451915158219708036, −20.01081988323226161089169907599, −19.53543229787408441038035704352, −18.068486022318266899384256702773, −17.695752399802832486352688312773, −16.79942418253759569680347634685, −16.3226878035440056950399471453, −14.98415162833964105985208687801, −13.99027132567682917283731330836, −13.0430763068817456389673024600, −12.07697907798067510192845971655, −11.22959830606531108985299188915, −9.98465327512965749686565138057, −9.74500428233350618260110834615, −8.6291485179092236433112948694, −7.62450818528869175629620190272, −6.83131935871148128842747701506, −5.66426826365199811740445989955, −4.38818267469940641592803261955, −3.20151914081361267861371590604, −1.72793410643547115428527986009, −1.106118959318297398625643416862, 1.23895389822199665832821891504, 2.300853229632559395789874884781, 3.27077058564806424187196993717, 5.2611906195856560275517390915, 5.985838986230963880747011514485, 6.74852094702800684431206977812, 7.97752190013467445953283952273, 8.7651195953270105418388953412, 9.71916462442475563277476720088, 10.31717548080836084803606512846, 11.571926264321623922036737196434, 12.02323893287121844189470168458, 13.71059194282326735978149468175, 14.44356429809597403903456923775, 15.18518118021529000527064951040, 16.25192372795753200050182126499, 17.00820997279698191190305362027, 17.883507137368622547316608419574, 18.68337078633053570609581197309, 19.005286591826068171215382806051, 20.35502440047361538371975562983, 21.115887330766789158499076139101, 21.95006036544883230423839368075, 22.806500379575376386900049165173, 24.10553009102661357819734017140

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