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Properties

Label 1-629-629.146-r0-0-0
Degree $1$
Conductor $629$
Sign $0.488 + 0.872i$
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  + (0.573 + 0.819i)2-s + (0.843 − 0.537i)3-s + (−0.342 + 0.939i)4-s + (0.887 + 0.461i)5-s + (0.923 + 0.382i)6-s + (0.953 − 0.300i)7-s + (−0.965 + 0.258i)8-s + (0.422 − 0.906i)9-s + (0.130 + 0.991i)10-s + (0.608 + 0.793i)11-s + (0.216 + 0.976i)12-s + (−0.939 − 0.342i)13-s + (0.793 + 0.608i)14-s + (0.996 − 0.0871i)15-s + (−0.766 − 0.642i)16-s + ⋯
L(s)  = 1  + (0.573 + 0.819i)2-s + (0.843 − 0.537i)3-s + (−0.342 + 0.939i)4-s + (0.887 + 0.461i)5-s + (0.923 + 0.382i)6-s + (0.953 − 0.300i)7-s + (−0.965 + 0.258i)8-s + (0.422 − 0.906i)9-s + (0.130 + 0.991i)10-s + (0.608 + 0.793i)11-s + (0.216 + 0.976i)12-s + (−0.939 − 0.342i)13-s + (0.793 + 0.608i)14-s + (0.996 − 0.0871i)15-s + (−0.766 − 0.642i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 629 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.488 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 629 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.488 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(629\)    =    \(17 \cdot 37\)
Sign: $0.488 + 0.872i$
Analytic conductor: \(2.92106\)
Root analytic conductor: \(2.92106\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{629} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 629,\ (0:\ ),\ 0.488 + 0.872i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.652179405 + 1.554486041i\)
\(L(\frac12)\) \(\approx\) \(2.652179405 + 1.554486041i\)
\(L(1)\) \(\approx\) \(1.975446702 + 0.7971851219i\)
\(L(1)\) \(\approx\) \(1.975446702 + 0.7971851219i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
37 \( 1 \)
good2 \( 1 + (0.573 + 0.819i)T \)
3 \( 1 + (0.843 - 0.537i)T \)
5 \( 1 + (0.887 + 0.461i)T \)
7 \( 1 + (0.953 - 0.300i)T \)
11 \( 1 + (0.608 + 0.793i)T \)
13 \( 1 + (-0.939 - 0.342i)T \)
19 \( 1 + (0.819 + 0.573i)T \)
23 \( 1 + (-0.130 - 0.991i)T \)
29 \( 1 + (-0.991 - 0.130i)T \)
31 \( 1 + (-0.923 + 0.382i)T \)
41 \( 1 + (0.737 + 0.675i)T \)
43 \( 1 + (-0.707 - 0.707i)T \)
47 \( 1 + (-0.866 + 0.5i)T \)
53 \( 1 + (-0.0871 - 0.996i)T \)
59 \( 1 + (0.0871 + 0.996i)T \)
61 \( 1 + (0.675 - 0.737i)T \)
67 \( 1 + (0.766 - 0.642i)T \)
71 \( 1 + (-0.843 + 0.537i)T \)
73 \( 1 + (-0.923 + 0.382i)T \)
79 \( 1 + (-0.953 + 0.300i)T \)
83 \( 1 + (-0.906 - 0.422i)T \)
89 \( 1 + (0.766 + 0.642i)T \)
97 \( 1 + (-0.608 + 0.793i)T \)
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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

−22.16640448576518137890057335764, −21.872765000183734328078466583711, −21.25233271398314426029878220571, −20.463495691342117404022434239354, −19.839141452756459400996736850876, −18.96765443717504535251363070205, −18.02972612098097668613716299465, −17.07069686040009579318500536539, −16.00709807560241217077651250283, −14.85602858947859436644034333384, −14.35974273811078534872665177650, −13.6766084932859111430346874456, −12.92639392561932864464466317832, −11.717045553509739265084956479214, −11.06675193212049757646448520106, −9.89243298511057559680278812207, −9.29459369058750335574403711899, −8.66406520654623955253754565633, −7.342516340231868546615081376471, −5.7403004172825990480448112302, −5.11961369622502518623599551944, −4.245893457621073244550346051308, −3.154992226169052444535286374887, −2.13507344767218588924479023916, −1.418160467281624109938386913267, 1.62086601511957916602782063088, 2.5209000050884781908529630643, 3.643962616819924685302923962400, 4.722867389773113910558629491547, 5.704665974425622792933662101, 6.864412716678444487528939789750, 7.359436238196950521422615534541, 8.22808502609673481804278826820, 9.278838012073651277543275945991, 10.04353002062849719748814782980, 11.54252935219153005645382918452, 12.51292741445063357400247350341, 13.205896968409167982416005226092, 14.35087361558807472686031589831, 14.43833944677269571264249129317, 15.10023766975787099146063695972, 16.52311043158538918753926328032, 17.453428734215592627498665911135, 17.93257956597386702607787861988, 18.68850519183602294866641672779, 20.13642385291354259225192277329, 20.62409881706322546008057362791, 21.55022033213549436591521506716, 22.362545867734721464879986369641, 23.131513502884374723693531863855

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