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Properties

Label 1-19e2-361.112-r0-0-0
Degree $1$
Conductor $361$
Sign $-0.988 - 0.153i$
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  + (−0.951 + 0.307i)2-s + (−0.842 − 0.539i)3-s + (0.811 − 0.584i)4-s + (−0.979 + 0.200i)5-s + (0.967 + 0.254i)6-s + (−0.962 − 0.272i)7-s + (−0.592 + 0.805i)8-s + (0.418 + 0.908i)9-s + (0.870 − 0.492i)10-s + (0.904 − 0.426i)11-s + (−0.998 + 0.0550i)12-s + (0.606 + 0.794i)13-s + (0.999 − 0.0367i)14-s + (0.933 + 0.359i)15-s + (0.315 − 0.948i)16-s + (0.100 − 0.994i)17-s + ⋯
L(s)  = 1  + (−0.951 + 0.307i)2-s + (−0.842 − 0.539i)3-s + (0.811 − 0.584i)4-s + (−0.979 + 0.200i)5-s + (0.967 + 0.254i)6-s + (−0.962 − 0.272i)7-s + (−0.592 + 0.805i)8-s + (0.418 + 0.908i)9-s + (0.870 − 0.492i)10-s + (0.904 − 0.426i)11-s + (−0.998 + 0.0550i)12-s + (0.606 + 0.794i)13-s + (0.999 − 0.0367i)14-s + (0.933 + 0.359i)15-s + (0.315 − 0.948i)16-s + (0.100 − 0.994i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.153i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.153i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(361\)    =    \(19^{2}\)
Sign: $-0.988 - 0.153i$
Analytic conductor: \(1.67647\)
Root analytic conductor: \(1.67647\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{361} (112, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 361,\ (0:\ ),\ -0.988 - 0.153i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.007162044240 - 0.09298503778i\)
\(L(\frac12)\) \(\approx\) \(0.007162044240 - 0.09298503778i\)
\(L(1)\) \(\approx\) \(0.3662855664 - 0.04307036813i\)
\(L(1)\) \(\approx\) \(0.3662855664 - 0.04307036813i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
good2 \( 1 + (-0.951 + 0.307i)T \)
3 \( 1 + (-0.842 - 0.539i)T \)
5 \( 1 + (-0.979 + 0.200i)T \)
7 \( 1 + (-0.962 - 0.272i)T \)
11 \( 1 + (0.904 - 0.426i)T \)
13 \( 1 + (0.606 + 0.794i)T \)
17 \( 1 + (0.100 - 0.994i)T \)
23 \( 1 + (-0.0459 - 0.998i)T \)
29 \( 1 + (-0.562 - 0.826i)T \)
31 \( 1 + (-0.926 + 0.376i)T \)
37 \( 1 + (-0.0825 + 0.996i)T \)
41 \( 1 + (0.484 + 0.875i)T \)
43 \( 1 + (-0.861 + 0.507i)T \)
47 \( 1 + (-0.263 - 0.964i)T \)
53 \( 1 + (-0.703 + 0.710i)T \)
59 \( 1 + (-0.999 - 0.0183i)T \)
61 \( 1 + (-0.896 + 0.443i)T \)
67 \( 1 + (0.690 + 0.723i)T \)
71 \( 1 + (-0.896 - 0.443i)T \)
73 \( 1 + (-0.912 + 0.410i)T \)
79 \( 1 + (-0.00918 - 0.999i)T \)
83 \( 1 + (0.0275 - 0.999i)T \)
89 \( 1 + (-0.912 - 0.410i)T \)
97 \( 1 + (0.690 - 0.723i)T \)
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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

−25.45390227232087605045145775495, −24.25279668086953148514260785811, −23.27399739438195916602467880417, −22.42673401440593249434858650718, −21.691883708013295303326890053902, −20.49986362617246807830063297118, −19.79606623894327553998205188920, −19.018608003110239576839909398, −18.012035627157662788816908623426, −17.09457238258942396299320902390, −16.36088915709958467528282289873, −15.61974859776408407155167059141, −14.99286790809753431448305003657, −12.74264432834215703213938633538, −12.38213038418813812439238013851, −11.30757138981938407454754749458, −10.656765336854053610886714792091, −9.548519274361300443354637388540, −8.87412596225514880242162910793, −7.59087617388209967201469943592, −6.617311207556927544381637185675, −5.63155830217632952574995063465, −3.87992494920450661586874113719, −3.42612794570014866560962948401, −1.38691511209183032215029403314, 0.09998603744981625635454292017, 1.3516752928538770922419207554, 3.03680231173192476156236800516, 4.45436155335312796272689465211, 6.056488428879776073292664845, 6.70194951369718307044109832996, 7.39641135197995469814178506278, 8.54168203404772665753946059403, 9.5825945583962545580982490956, 10.73156659447124629843713399228, 11.51665381859588241231255120147, 12.10502797280881193541756814044, 13.45255696733690157978754353407, 14.62067269913763791826792516446, 15.91242118933715003771813692548, 16.41899834633576818305193203483, 16.97945512164658894040818433483, 18.41683576978499326122690119192, 18.7569939035735806459501586612, 19.5834309235119007394311785002, 20.35792442540300345039987346485, 21.90839646286123816420328262706, 22.916660873420776898010864068410, 23.415887446192466028504775459, 24.39508845200448378065072542502

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