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Properties

Label 1-507-507.368-r1-0-0
Degree $1$
Conductor $507$
Sign $0.963 + 0.269i$
Analytic cond. $54.4847$
Root an. cond. $54.4847$
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  + (−0.632 + 0.774i)2-s + (−0.200 − 0.979i)4-s + (0.120 + 0.992i)5-s + (0.845 − 0.534i)7-s + (0.885 + 0.464i)8-s + (−0.845 − 0.534i)10-s + (0.987 − 0.160i)11-s + (−0.120 + 0.992i)14-s + (−0.919 + 0.391i)16-s + (0.845 − 0.534i)17-s + (0.5 − 0.866i)19-s + (0.948 − 0.316i)20-s + (−0.5 + 0.866i)22-s + (0.5 + 0.866i)23-s + (−0.970 + 0.239i)25-s + ⋯
L(s)  = 1  + (−0.632 + 0.774i)2-s + (−0.200 − 0.979i)4-s + (0.120 + 0.992i)5-s + (0.845 − 0.534i)7-s + (0.885 + 0.464i)8-s + (−0.845 − 0.534i)10-s + (0.987 − 0.160i)11-s + (−0.120 + 0.992i)14-s + (−0.919 + 0.391i)16-s + (0.845 − 0.534i)17-s + (0.5 − 0.866i)19-s + (0.948 − 0.316i)20-s + (−0.5 + 0.866i)22-s + (0.5 + 0.866i)23-s + (−0.970 + 0.239i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.963 + 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.963 + 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.963 + 0.269i$
Analytic conductor: \(54.4847\)
Root analytic conductor: \(54.4847\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (368, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 507,\ (1:\ ),\ 0.963 + 0.269i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.816724190 + 0.2492043204i\)
\(L(\frac12)\) \(\approx\) \(1.816724190 + 0.2492043204i\)
\(L(1)\) \(\approx\) \(0.9743518556 + 0.2714769166i\)
\(L(1)\) \(\approx\) \(0.9743518556 + 0.2714769166i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.632 + 0.774i)T \)
5 \( 1 + (0.120 + 0.992i)T \)
7 \( 1 + (0.845 - 0.534i)T \)
11 \( 1 + (0.987 - 0.160i)T \)
17 \( 1 + (0.845 - 0.534i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (0.632 - 0.774i)T \)
31 \( 1 + (0.970 + 0.239i)T \)
37 \( 1 + (-0.278 - 0.960i)T \)
41 \( 1 + (-0.996 - 0.0804i)T \)
43 \( 1 + (0.278 - 0.960i)T \)
47 \( 1 + (-0.748 - 0.663i)T \)
53 \( 1 + (-0.885 - 0.464i)T \)
59 \( 1 + (-0.919 - 0.391i)T \)
61 \( 1 + (-0.0402 + 0.999i)T \)
67 \( 1 + (0.200 - 0.979i)T \)
71 \( 1 + (0.428 + 0.903i)T \)
73 \( 1 + (0.354 - 0.935i)T \)
79 \( 1 + (-0.748 - 0.663i)T \)
83 \( 1 + (0.568 + 0.822i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (-0.799 - 0.600i)T \)
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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

−23.29823848522654957327024134536, −22.24499618606644788830740431219, −21.39891248175088099589006222365, −20.76440812859951596744039802493, −20.121309245620560611035036201506, −19.14360879252107220220573422447, −18.38233835587770115078931027158, −17.35206007456479950671714755184, −16.8996773217786442271894178784, −15.97590964275448815266533478555, −14.67274558281659435438877216152, −13.79733162537185089950636028892, −12.49707423578468970583463692600, −12.16631789365415274812000798649, −11.28908392672571323939312222439, −10.11893335054322439206190995570, −9.31931677060031814883757781125, −8.413206009212357414279716813921, −7.93585729848287396507244221964, −6.42208401177700680374056769128, −5.07081281495808624168011071562, −4.274557835180986656729865017605, −3.03902509539631284182265774494, −1.59308119335842796953200370162, −1.13435751761983613461775425520, 0.69876172244786194475551979142, 1.81067736869320586225510067940, 3.32161625723785541076442883313, 4.63395947529683695112243723367, 5.661827416758429173748457993088, 6.78303089020055200284925951235, 7.321645100692214186934634053, 8.27434757050689108001771678351, 9.39550749335499762560453133428, 10.17365026786404741894723767830, 11.13814447831365166498696998286, 11.77228872097738210905558949633, 13.7928180535895556152335042784, 13.97299798086211753064478489130, 14.95435999820806397333043322005, 15.66167319010559142331858229388, 16.86415206081567613815426708311, 17.499070314014294036162088163756, 18.136431695200420659772561617396, 19.137022500513234945550897826594, 19.71519640539764230764393786735, 20.87959903239467504108202189832, 21.90088949986960424191472426073, 22.88632129302344173228573764438, 23.435165469140783123780649433754

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