Properties

Degree 1
Conductor $ 13 \cdot 31 $
Sign $0.780 + 0.624i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.207 + 0.978i)2-s + (0.978 − 0.207i)3-s + (−0.913 − 0.406i)4-s + (−0.866 + 0.5i)5-s + i·6-s + (−0.587 − 0.809i)7-s + (0.587 − 0.809i)8-s + (0.913 − 0.406i)9-s + (−0.309 − 0.951i)10-s + (0.587 + 0.809i)11-s + (−0.978 − 0.207i)12-s + (0.913 − 0.406i)14-s + (−0.743 + 0.669i)15-s + (0.669 + 0.743i)16-s + (−0.809 − 0.587i)17-s + (0.207 + 0.978i)18-s + ⋯
L(s,χ)  = 1  + (−0.207 + 0.978i)2-s + (0.978 − 0.207i)3-s + (−0.913 − 0.406i)4-s + (−0.866 + 0.5i)5-s + i·6-s + (−0.587 − 0.809i)7-s + (0.587 − 0.809i)8-s + (0.913 − 0.406i)9-s + (−0.309 − 0.951i)10-s + (0.587 + 0.809i)11-s + (−0.978 − 0.207i)12-s + (0.913 − 0.406i)14-s + (−0.743 + 0.669i)15-s + (0.669 + 0.743i)16-s + (−0.809 − 0.587i)17-s + (0.207 + 0.978i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.780 + 0.624i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.780 + 0.624i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(403\)    =    \(13 \cdot 31\)
\( \varepsilon \)  =  $0.780 + 0.624i$
motivic weight  =  \(0\)
character  :  $\chi_{403} (353, \cdot )$
Sato-Tate  :  $\mu(60)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 403,\ (0:\ ),\ 0.780 + 0.624i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.214080482 + 0.4258315577i$
$L(\frac12,\chi)$  $\approx$  $1.214080482 + 0.4258315577i$
$L(\chi,1)$  $\approx$  1.023970076 + 0.3430646601i
$L(1,\chi)$  $\approx$  1.023970076 + 0.3430646601i

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−24.44627474527467681636568252203, −23.233098463768899053926555923538, −22.17005447326921170232661960939, −21.57688265893610946800851822377, −20.614461418165362364277811053675, −19.83443637084000260561923577020, −19.192731240592413139912127123981, −18.76286085409148460452469024378, −17.39328210570101437239060617193, −16.1812042946336334081738963198, −15.53157988760012254072869235554, −14.386274854066843045813321958280, −13.42599142338856095506768882882, −12.60964964886910819958675691014, −11.80791548977071755727487600126, −10.82482912890046467511480547782, −9.60968303276169353645298752309, −8.82064946808366735794394190189, −8.421406218239730043019620684567, −7.1741868998303013264196948033, −5.42560592518387268693257851134, −4.16146719107063105398219391961, −3.451711892325939475290401934869, −2.54539884123167302528746226462, −1.13456518814163344857315857068, 0.96547399253888529547361723912, 2.87282364654017347359144008356, 3.947578154372122713119827161768, 4.70087351335444755594237869900, 6.72020378725116480292903144820, 6.99477293651143544003844359201, 7.82312204438478467034822465807, 8.92386665466515198812646564524, 9.683360726472329840117566474759, 10.69856018462897310453642450026, 12.2038471236904672984583235374, 13.244711614787095881663036858996, 14.0519783273419403092047686686, 14.784083880079596930770551501461, 15.63919275882402428513684482452, 16.23498020853476705175183177859, 17.5336317469024890441569048338, 18.29742568553610379562888313002, 19.42262041530592038450840831554, 19.68788971147688685169549029587, 20.71006805180937051034798091749, 22.38873967910353929966986381376, 22.75273300149469173351535012706, 23.77528216357504334208427165988, 24.50733899244926654199081713416

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