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Properties

Label 1-19e2-361.149-r0-0-0
Degree $1$
Conductor $361$
Sign $0.970 + 0.239i$
Analytic cond. $1.67647$
Root an. cond. $1.67647$
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  + (−0.00918 − 0.999i)2-s + (0.957 + 0.289i)3-s + (−0.999 + 0.0183i)4-s + (−0.531 − 0.847i)5-s + (0.280 − 0.959i)6-s + (−0.191 + 0.981i)7-s + (0.0275 + 0.999i)8-s + (0.832 + 0.554i)9-s + (−0.842 + 0.539i)10-s + (−0.592 + 0.805i)11-s + (−0.962 − 0.272i)12-s + (−0.119 + 0.992i)13-s + (0.983 + 0.182i)14-s + (−0.263 − 0.964i)15-s + (0.999 − 0.0367i)16-s + (0.484 + 0.875i)17-s + ⋯
L(s)  = 1  + (−0.00918 − 0.999i)2-s + (0.957 + 0.289i)3-s + (−0.999 + 0.0183i)4-s + (−0.531 − 0.847i)5-s + (0.280 − 0.959i)6-s + (−0.191 + 0.981i)7-s + (0.0275 + 0.999i)8-s + (0.832 + 0.554i)9-s + (−0.842 + 0.539i)10-s + (−0.592 + 0.805i)11-s + (−0.962 − 0.272i)12-s + (−0.119 + 0.992i)13-s + (0.983 + 0.182i)14-s + (−0.263 − 0.964i)15-s + (0.999 − 0.0367i)16-s + (0.484 + 0.875i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.239i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.239i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(361\)    =    \(19^{2}\)
Sign: $0.970 + 0.239i$
Analytic conductor: \(1.67647\)
Root analytic conductor: \(1.67647\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{361} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 361,\ (0:\ ),\ 0.970 + 0.239i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.232653627 + 0.1497177666i\)
\(L(\frac12)\) \(\approx\) \(1.232653627 + 0.1497177666i\)
\(L(1)\) \(\approx\) \(1.092195288 - 0.1881902917i\)
\(L(1)\) \(\approx\) \(1.092195288 - 0.1881902917i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
good2 \( 1 + (-0.00918 - 0.999i)T \)
3 \( 1 + (0.957 + 0.289i)T \)
5 \( 1 + (-0.531 - 0.847i)T \)
7 \( 1 + (-0.191 + 0.981i)T \)
11 \( 1 + (-0.592 + 0.805i)T \)
13 \( 1 + (-0.119 + 0.992i)T \)
17 \( 1 + (0.484 + 0.875i)T \)
23 \( 1 + (-0.227 + 0.973i)T \)
29 \( 1 + (-0.155 - 0.987i)T \)
31 \( 1 + (0.350 - 0.936i)T \)
37 \( 1 + (-0.401 - 0.915i)T \)
41 \( 1 + (0.577 + 0.816i)T \)
43 \( 1 + (0.888 - 0.459i)T \)
47 \( 1 + (-0.971 + 0.236i)T \)
53 \( 1 + (0.690 + 0.723i)T \)
59 \( 1 + (-0.995 + 0.0917i)T \)
61 \( 1 + (0.663 - 0.748i)T \)
67 \( 1 + (-0.621 + 0.783i)T \)
71 \( 1 + (0.663 + 0.748i)T \)
73 \( 1 + (0.515 - 0.856i)T \)
79 \( 1 + (-0.0459 + 0.998i)T \)
83 \( 1 + (0.137 + 0.990i)T \)
89 \( 1 + (0.515 + 0.856i)T \)
97 \( 1 + (-0.621 - 0.783i)T \)
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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

−24.6536942074860468637544758834, −23.96720545532969736054257488652, −23.07899130733803121051908883714, −22.4419719618595705117936534710, −21.17289373397624167078774100369, −20.09736330978949035464398819666, −19.23975902074561214984015585653, −18.453248861639208736359894655001, −17.77289709956930963000399019996, −16.348137970644004025025572155740, −15.76600762384301048037136713214, −14.718649721651790145798452317839, −14.11026645670353151994031291709, −13.37616819190696197541597241536, −12.38678488766389333837897589247, −10.65759706700867629707409684352, −10.00461704678811542706965399156, −8.65062554005326122781843532443, −7.82143969282924702912543036098, −7.23883920342817482738572569915, −6.34165967504376662322414701592, −4.85673932795252590662370671661, −3.56617115261197391084988827349, −2.97210840725448383383745728305, −0.711347323707314405359465727400, 1.66034129157294565876390001634, 2.47675603862176303711011558334, 3.78262369747392208940501447144, 4.50290304593486333025333757089, 5.59857851755426212071364398141, 7.6562992335401783742150689841, 8.37929485095530706120746086904, 9.36820479938192600676128472743, 9.777761530118121455430244047519, 11.22345163768847990510533360738, 12.252531175679526218502131111413, 12.81733693635831592852821264253, 13.7757665644364253425664784224, 14.93152313636483325696385652048, 15.60233168337173942192168309042, 16.75206060303610602806891341484, 17.97013873868930187434110817143, 19.197605922912980922121925122904, 19.30543888440386743292132595408, 20.45012464153715688520259923463, 21.157219640680762834630757702952, 21.64897775847722467837333913010, 22.88863018593920776914509886808, 23.85209972351621516806402426004, 24.75628164526542308602781336354

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