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Properties

Label 1-403-403.353-r0-0-0
Degree $1$
Conductor $403$
Sign $0.780 + 0.624i$
Analytic cond. $1.87152$
Root an. cond. $1.87152$
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  + (−0.207 + 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.780 + 0.624i$
Analytic conductor: \(1.87152\)
Root analytic conductor: \(1.87152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 403,\ (0:\ ),\ 0.780 + 0.624i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.214080482 + 0.4258315577i\)
\(L(\frac12)\) \(\approx\) \(1.214080482 + 0.4258315577i\)
\(L(1)\) \(\approx\) \(1.023970076 + 0.3430646601i\)
\(L(1)\) \(\approx\) \(1.023970076 + 0.3430646601i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.207 + 0.978i)T \)
3 \( 1 + (0.978 - 0.207i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (-0.587 - 0.809i)T \)
11 \( 1 + (0.587 + 0.809i)T \)
17 \( 1 + (-0.809 - 0.587i)T \)
19 \( 1 + (0.951 - 0.309i)T \)
23 \( 1 + (0.913 - 0.406i)T \)
29 \( 1 + (0.978 + 0.207i)T \)
37 \( 1 + (-0.866 + 0.5i)T \)
41 \( 1 + (0.951 - 0.309i)T \)
43 \( 1 + (0.309 + 0.951i)T \)
47 \( 1 + (0.951 + 0.309i)T \)
53 \( 1 + (0.104 - 0.994i)T \)
59 \( 1 + (0.951 + 0.309i)T \)
61 \( 1 + (0.5 + 0.866i)T \)
67 \( 1 - iT \)
71 \( 1 + (-0.406 - 0.913i)T \)
73 \( 1 + (0.406 - 0.913i)T \)
79 \( 1 + (-0.913 + 0.406i)T \)
83 \( 1 + (0.743 + 0.669i)T \)
89 \( 1 + (-0.994 + 0.104i)T \)
97 \( 1 + (0.406 - 0.913i)T \)
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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

−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