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Properties

Label 1-403-403.211-r0-0-0
Degree $1$
Conductor $403$
Sign $0.929 - 0.368i$
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  + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.5 − 0.866i)4-s + (−0.5 − 0.866i)5-s + 6-s + 7-s + 8-s + (−0.5 + 0.866i)9-s + 10-s + 11-s + (−0.5 + 0.866i)12-s + (−0.5 + 0.866i)14-s + (−0.5 + 0.866i)15-s + (−0.5 + 0.866i)16-s + 17-s + (−0.5 − 0.866i)18-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.5 − 0.866i)4-s + (−0.5 − 0.866i)5-s + 6-s + 7-s + 8-s + (−0.5 + 0.866i)9-s + 10-s + 11-s + (−0.5 + 0.866i)12-s + (−0.5 + 0.866i)14-s + (−0.5 + 0.866i)15-s + (−0.5 + 0.866i)16-s + 17-s + (−0.5 − 0.866i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.929 - 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.929 - 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.929 - 0.368i$
Analytic conductor: \(1.87152\)
Root analytic conductor: \(1.87152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 403,\ (0:\ ),\ 0.929 - 0.368i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8847224533 - 0.1691689094i\)
\(L(\frac12)\) \(\approx\) \(0.8847224533 - 0.1691689094i\)
\(L(1)\) \(\approx\) \(0.7805416416 - 0.03625445636i\)
\(L(1)\) \(\approx\) \(0.7805416416 - 0.03625445636i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + T \)
11 \( 1 + T \)
17 \( 1 + T \)
19 \( 1 + T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + T \)
43 \( 1 + T \)
47 \( 1 + T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + T \)
71 \( 1 + (-0.5 + 0.866i)T \)
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

−24.327177969359914300658975177544, −23.10476436449895950288015140032, −22.43400544749358879157828250928, −21.845037234064837000646859037, −20.84855972732194021936835492439, −20.27024704239796481658144589424, −19.15067984312762663293486114746, −18.31494950163523026348776675373, −17.50550508129891820536948667590, −16.75166003543751810909944294329, −15.720775674222909055517853804753, −14.55996686672905859794808239449, −14.04666424095745281950237812943, −12.147693568415667363025296260014, −11.73435605501912532399213892280, −10.95484189146921188827095981510, −10.18499764022580823742282785313, −9.28787321570502222038513649639, −8.182373836982065083046428176397, −7.23176228794710874457209185302, −5.78774011910077924837439495777, −4.43087565475953049541950434845, −3.78355942225480550328241031056, −2.67244188028637885874146802079, −1.07361895305273691432943458450, 0.96878510190201682482294987520, 1.641227731048053291020181168072, 4.00252950113873816352617352121, 5.229697298690938685994346129563, 5.73118610654433869640875082692, 7.20176945443742191002042342809, 7.705586393275866613971378658861, 8.59810051802033896016748621661, 9.52772655587826180480022663861, 11.01046569703415955618722644253, 11.78452943185868665187463455101, 12.67039963382176395582761975848, 13.9509792701791443474459570333, 14.449664950377928606782610760522, 15.80066881887721147379116131952, 16.56490189097513957267371558966, 17.34713592927311029658442532078, 17.916561333608548040820538464949, 18.957859354776103503447929547314, 19.68218099458472944798574110509, 20.56072975179844054732042948543, 21.99234987067715277554937386675, 23.00995965093261702534575116159, 23.72292367070727140581882852657, 24.39809702995557170554610192817

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