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Properties

Label 1-629-629.190-r0-0-0
Degree $1$
Conductor $629$
Sign $0.561 + 0.827i$
Analytic cond. $2.92106$
Root an. cond. $2.92106$
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  + (−0.996 − 0.0871i)2-s + (−0.461 − 0.887i)3-s + (0.984 + 0.173i)4-s + (0.999 + 0.0436i)5-s + (0.382 + 0.923i)6-s + (0.675 + 0.737i)7-s + (−0.965 − 0.258i)8-s + (−0.573 + 0.819i)9-s + (−0.991 − 0.130i)10-s + (−0.793 − 0.608i)11-s + (−0.300 − 0.953i)12-s + (−0.173 + 0.984i)13-s + (−0.608 − 0.793i)14-s + (−0.422 − 0.906i)15-s + (0.939 + 0.342i)16-s + ⋯
L(s)  = 1  + (−0.996 − 0.0871i)2-s + (−0.461 − 0.887i)3-s + (0.984 + 0.173i)4-s + (0.999 + 0.0436i)5-s + (0.382 + 0.923i)6-s + (0.675 + 0.737i)7-s + (−0.965 − 0.258i)8-s + (−0.573 + 0.819i)9-s + (−0.991 − 0.130i)10-s + (−0.793 − 0.608i)11-s + (−0.300 − 0.953i)12-s + (−0.173 + 0.984i)13-s + (−0.608 − 0.793i)14-s + (−0.422 − 0.906i)15-s + (0.939 + 0.342i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 629 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.561 + 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 629 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.561 + 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(629\)    =    \(17 \cdot 37\)
Sign: $0.561 + 0.827i$
Analytic conductor: \(2.92106\)
Root analytic conductor: \(2.92106\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{629} (190, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 629,\ (0:\ ),\ 0.561 + 0.827i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6252680361 + 0.3314713270i\)
\(L(\frac12)\) \(\approx\) \(0.6252680361 + 0.3314713270i\)
\(L(1)\) \(\approx\) \(0.6727012250 + 0.005765310776i\)
\(L(1)\) \(\approx\) \(0.6727012250 + 0.005765310776i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
37 \( 1 \)
good2 \( 1 + (-0.996 - 0.0871i)T \)
3 \( 1 + (-0.461 - 0.887i)T \)
5 \( 1 + (0.999 + 0.0436i)T \)
7 \( 1 + (0.675 + 0.737i)T \)
11 \( 1 + (-0.793 - 0.608i)T \)
13 \( 1 + (-0.173 + 0.984i)T \)
19 \( 1 + (0.0871 + 0.996i)T \)
23 \( 1 + (-0.991 - 0.130i)T \)
29 \( 1 + (0.130 + 0.991i)T \)
31 \( 1 + (-0.382 + 0.923i)T \)
41 \( 1 + (0.976 - 0.216i)T \)
43 \( 1 + (-0.707 + 0.707i)T \)
47 \( 1 + (-0.866 - 0.5i)T \)
53 \( 1 + (-0.906 + 0.422i)T \)
59 \( 1 + (-0.906 + 0.422i)T \)
61 \( 1 + (-0.216 - 0.976i)T \)
67 \( 1 + (-0.939 + 0.342i)T \)
71 \( 1 + (0.461 + 0.887i)T \)
73 \( 1 + (0.382 - 0.923i)T \)
79 \( 1 + (0.675 + 0.737i)T \)
83 \( 1 + (-0.819 - 0.573i)T \)
89 \( 1 + (0.939 + 0.342i)T \)
97 \( 1 + (-0.793 + 0.608i)T \)
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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

−22.772942416187739788458537479134, −21.801904309511759500799238596359, −20.92647976357638760041658166782, −20.51231645109396388406886975827, −19.75756133920066770018906992626, −18.18238274776169446330641682469, −17.75180582071213884382836111716, −17.24294432328245213625050813894, −16.410141247337515386046375282096, −15.44156606317492143263503222452, −14.843671287711296482772136979373, −13.7202555145799125111156962868, −12.580511378605500238627169512269, −11.37231546451002482162411312930, −10.683268558027856712160197894818, −9.99676388609365833306330734315, −9.49170965240523384255267060059, −8.27531472504920803628825013936, −7.43757346947020932392346949201, −6.24865817131102650561593454365, −5.42374659507347103639368803120, −4.537701382723993635926061663957, −2.99101974704613786576319165775, −1.93475787958863247255294985976, −0.503608520864796483122401664948, 1.44436732485497556338951908421, 1.97671290315768557281779676632, 2.95691202873698763337933264221, 5.091967862042456129526776985754, 5.91497051191458988533570887096, 6.58903697214653418139239036997, 7.72972297322446640038965156024, 8.442527304641892050833255085860, 9.31672046603560759085050244929, 10.43387144063818241537190262945, 11.11946939690840290661484003787, 12.05720872171145434302849393330, 12.704565962916347212668196245937, 13.94937175550359479185120226066, 14.58073717758705683023804154974, 16.08556373514156050471840949182, 16.61053795782704741765197767980, 17.58449107872126938604929130366, 18.30721627754346958285200170120, 18.50708128106102933957339710206, 19.50103229145544054620559686735, 20.5959574788371903466086441017, 21.48896176512325540683180807729, 21.88261922028447045315865258272, 23.36476874057196140606579227961

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