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Properties

Label 1-19e2-361.150-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: $\chi_{361} (150, \cdot )$
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

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

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