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Properties

Label 1-19e2-361.157-r0-0-0
Degree $1$
Conductor $361$
Sign $-0.911 - 0.411i$
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  + (−0.896 − 0.443i)2-s + (−0.531 − 0.847i)3-s + (0.606 + 0.794i)4-s + (0.967 − 0.254i)5-s + (0.100 + 0.994i)6-s + (0.975 − 0.218i)7-s + (−0.191 − 0.981i)8-s + (−0.435 + 0.900i)9-s + (−0.979 − 0.200i)10-s + (−0.962 + 0.272i)11-s + (0.350 − 0.936i)12-s + (−0.951 − 0.307i)13-s + (−0.971 − 0.236i)14-s + (−0.729 − 0.684i)15-s + (−0.263 + 0.964i)16-s + (−0.991 − 0.128i)17-s + ⋯
L(s)  = 1  + (−0.896 − 0.443i)2-s + (−0.531 − 0.847i)3-s + (0.606 + 0.794i)4-s + (0.967 − 0.254i)5-s + (0.100 + 0.994i)6-s + (0.975 − 0.218i)7-s + (−0.191 − 0.981i)8-s + (−0.435 + 0.900i)9-s + (−0.979 − 0.200i)10-s + (−0.962 + 0.272i)11-s + (0.350 − 0.936i)12-s + (−0.951 − 0.307i)13-s + (−0.971 − 0.236i)14-s + (−0.729 − 0.684i)15-s + (−0.263 + 0.964i)16-s + (−0.991 − 0.128i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.911 - 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.911 - 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(361\)    =    \(19^{2}\)
Sign: $-0.911 - 0.411i$
Analytic conductor: \(1.67647\)
Root analytic conductor: \(1.67647\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{361} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 361,\ (0:\ ),\ -0.911 - 0.411i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1300043482 - 0.6032988435i\)
\(L(\frac12)\) \(\approx\) \(0.1300043482 - 0.6032988435i\)
\(L(1)\) \(\approx\) \(0.5223555877 - 0.3774146763i\)
\(L(1)\) \(\approx\) \(0.5223555877 - 0.3774146763i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
good2 \( 1 + (-0.896 - 0.443i)T \)
3 \( 1 + (-0.531 - 0.847i)T \)
5 \( 1 + (0.967 - 0.254i)T \)
7 \( 1 + (0.975 - 0.218i)T \)
11 \( 1 + (-0.962 + 0.272i)T \)
13 \( 1 + (-0.951 - 0.307i)T \)
17 \( 1 + (-0.991 - 0.128i)T \)
23 \( 1 + (-0.467 - 0.883i)T \)
29 \( 1 + (-0.0459 - 0.998i)T \)
31 \( 1 + (-0.592 - 0.805i)T \)
37 \( 1 + (0.245 - 0.969i)T \)
41 \( 1 + (-0.800 + 0.599i)T \)
43 \( 1 + (0.315 - 0.948i)T \)
47 \( 1 + (0.811 - 0.584i)T \)
53 \( 1 + (-0.912 - 0.410i)T \)
59 \( 1 + (-0.119 - 0.992i)T \)
61 \( 1 + (-0.155 - 0.987i)T \)
67 \( 1 + (0.515 + 0.856i)T \)
71 \( 1 + (-0.155 + 0.987i)T \)
73 \( 1 + (0.384 + 0.922i)T \)
79 \( 1 + (0.663 + 0.748i)T \)
83 \( 1 + (-0.821 - 0.569i)T \)
89 \( 1 + (0.384 - 0.922i)T \)
97 \( 1 + (0.515 - 0.856i)T \)
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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

−25.34628234365937870100118356778, −24.136038978705362236257402312443, −23.74842244674089706258245971087, −22.24742839154457324189124907963, −21.54806897849083369477758898262, −20.80356250527343623293679950676, −19.874252972266119015869143581699, −18.46596125208063757398250899090, −17.829792825883973368107009018514, −17.27659281292708308364297204225, −16.36011521117596886472461975674, −15.37545469694028795055478001579, −14.69076428211428394229217415482, −13.76078451631126425460932632021, −12.10867193510285607524396376342, −10.96532267081775971186804613962, −10.53871035611512427187078343041, −9.50181802563809193757076048568, −8.78899958907541449283936484411, −7.546233275831839463183598730487, −6.3924305431045079189962642192, −5.39541742423643746740594884871, −4.820799001556901357066477051792, −2.76868308838466217892120829788, −1.62211465763051233757476857241, 0.509866902389510863025764433418, 2.083802547570666454300970993838, 2.31094571545574622687143319201, 4.59996909325958628506917019639, 5.65120622943131502126613268604, 6.86474857804359822777743062475, 7.746097202468547301411796161142, 8.54285729168599472313647647891, 9.839354892630697763623868354163, 10.65508587423944060823562565437, 11.47877443194805389306208100843, 12.56678742876108856084460820952, 13.17081449811104868782705954091, 14.28045803501858229706362988730, 15.66971759886122647480179475865, 16.9961155469234971041149837926, 17.32954366796401219688195435718, 18.14971177453917972866837990379, 18.66647977678714648380204087584, 20.04518503084313911786407330727, 20.565562384308474576705944006404, 21.64797818825324954878992842070, 22.355094100636213336885709820827, 23.77472997798737626239436391616, 24.57980829954886313284702798003

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