Properties

Label 1-19e2-361.142-r0-0-0
Degree $1$
Conductor $361$
Sign $-0.204 + 0.978i$
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  + (0.100 + 0.994i)2-s + (−0.995 − 0.0917i)3-s + (−0.979 + 0.200i)4-s + (−0.119 + 0.992i)5-s + (−0.00918 − 0.999i)6-s + (0.851 + 0.523i)7-s + (−0.298 − 0.954i)8-s + (0.983 + 0.182i)9-s + (−0.999 − 0.0183i)10-s + (0.635 − 0.771i)11-s + (0.993 − 0.110i)12-s + (0.967 − 0.254i)13-s + (−0.435 + 0.900i)14-s + (0.209 − 0.977i)15-s + (0.919 − 0.393i)16-s + (0.663 − 0.748i)17-s + ⋯
L(s)  = 1  + (0.100 + 0.994i)2-s + (−0.995 − 0.0917i)3-s + (−0.979 + 0.200i)4-s + (−0.119 + 0.992i)5-s + (−0.00918 − 0.999i)6-s + (0.851 + 0.523i)7-s + (−0.298 − 0.954i)8-s + (0.983 + 0.182i)9-s + (−0.999 − 0.0183i)10-s + (0.635 − 0.771i)11-s + (0.993 − 0.110i)12-s + (0.967 − 0.254i)13-s + (−0.435 + 0.900i)14-s + (0.209 − 0.977i)15-s + (0.919 − 0.393i)16-s + (0.663 − 0.748i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.204 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.204 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(361\)    =    \(19^{2}\)
Sign: $-0.204 + 0.978i$
Analytic conductor: \(1.67647\)
Root analytic conductor: \(1.67647\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{361} (142, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 361,\ (0:\ ),\ -0.204 + 0.978i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6689094318 + 0.8235076602i\)
\(L(\frac12)\) \(\approx\) \(0.6689094318 + 0.8235076602i\)
\(L(1)\) \(\approx\) \(0.7436726160 + 0.5336618786i\)
\(L(1)\) \(\approx\) \(0.7436726160 + 0.5336618786i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
good2 \( 1 + (0.100 + 0.994i)T \)
3 \( 1 + (-0.995 - 0.0917i)T \)
5 \( 1 + (-0.119 + 0.992i)T \)
7 \( 1 + (0.851 + 0.523i)T \)
11 \( 1 + (0.635 - 0.771i)T \)
13 \( 1 + (0.967 - 0.254i)T \)
17 \( 1 + (0.663 - 0.748i)T \)
23 \( 1 + (0.577 + 0.816i)T \)
29 \( 1 + (0.989 - 0.146i)T \)
31 \( 1 + (0.716 - 0.697i)T \)
37 \( 1 + (-0.986 - 0.164i)T \)
41 \( 1 + (-0.467 - 0.883i)T \)
43 \( 1 + (0.515 + 0.856i)T \)
47 \( 1 + (0.870 + 0.492i)T \)
53 \( 1 + (-0.861 + 0.507i)T \)
59 \( 1 + (-0.531 + 0.847i)T \)
61 \( 1 + (-0.991 - 0.128i)T \)
67 \( 1 + (0.888 - 0.459i)T \)
71 \( 1 + (-0.991 + 0.128i)T \)
73 \( 1 + (0.315 + 0.948i)T \)
79 \( 1 + (0.484 - 0.875i)T \)
83 \( 1 + (-0.998 - 0.0550i)T \)
89 \( 1 + (0.315 - 0.948i)T \)
97 \( 1 + (0.888 + 0.459i)T \)
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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

−24.03499913876644059791352795971, −23.43242841901307800974654195233, −22.82989242768319965078897185313, −21.61446105055991218001145994435, −20.96130065206458013734291807298, −20.35312248687509822226527810941, −19.27134246935630847473119877634, −18.22934810593197449722018345616, −17.30134328662351794884786199573, −16.93049202757762897825947807303, −15.6315708206473185994904841888, −14.38985232047116352245289621408, −13.373904053518477136661726950318, −12.34147822302782457521041565216, −11.92518331095327730568763209221, −10.85665519265554109178564780543, −10.171158345814664863943004446496, −8.98401961145279463309143453424, −8.0734290621801845956361085529, −6.53723902597651403118036715261, −5.223471617671763729036519996726, −4.5489830962509026971819462294, −3.78192613398885714241014491692, −1.62415275365462984629243910908, −1.05045625355897701169906487325, 1.138683696656605325155665734859, 3.20629724839653257136571272980, 4.41636360479281384390377882346, 5.60320339952461347453046891783, 6.15658298621761676599046118060, 7.1816666473027738397951894337, 8.06007157316010220515037704480, 9.24055235040778712444373989667, 10.51478663706647859056177066001, 11.412785693293541650363762032249, 12.168263634937307132385233303863, 13.62170859101925064456039596532, 14.22501104751214426657744904450, 15.41824325929315907219289563661, 15.88496326085982600139380782339, 17.11670197775526076508145222755, 17.72006132524416790237416775430, 18.60964235231119009943872399968, 19.02723305942510347536096899044, 21.1078701447634842425459135044, 21.71838681387394601370715939076, 22.655509277146041537826556468188, 23.15422121260223470168140455781, 24.06289242140015036390747453076, 24.85919360901765752157396588749

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