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Properties

Label 1-19e2-361.137-r0-0-0
Degree $1$
Conductor $361$
Sign $0.999 + 0.0174i$
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  + (0.957 − 0.289i)2-s + (−0.999 + 0.0183i)3-s + (0.832 − 0.554i)4-s + (0.606 + 0.794i)5-s + (−0.951 + 0.307i)6-s + (0.993 − 0.110i)7-s + (0.635 − 0.771i)8-s + (0.999 − 0.0367i)9-s + (0.811 + 0.584i)10-s + (0.137 + 0.990i)11-s + (−0.821 + 0.569i)12-s + (−0.777 − 0.628i)13-s + (0.919 − 0.393i)14-s + (−0.621 − 0.783i)15-s + (0.384 − 0.922i)16-s + (−0.896 + 0.443i)17-s + ⋯
L(s)  = 1  + (0.957 − 0.289i)2-s + (−0.999 + 0.0183i)3-s + (0.832 − 0.554i)4-s + (0.606 + 0.794i)5-s + (−0.951 + 0.307i)6-s + (0.993 − 0.110i)7-s + (0.635 − 0.771i)8-s + (0.999 − 0.0367i)9-s + (0.811 + 0.584i)10-s + (0.137 + 0.990i)11-s + (−0.821 + 0.569i)12-s + (−0.777 − 0.628i)13-s + (0.919 − 0.393i)14-s + (−0.621 − 0.783i)15-s + (0.384 − 0.922i)16-s + (−0.896 + 0.443i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0174i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0174i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(361\)    =    \(19^{2}\)
Sign: $0.999 + 0.0174i$
Analytic conductor: \(1.67647\)
Root analytic conductor: \(1.67647\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{361} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 361,\ (0:\ ),\ 0.999 + 0.0174i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.100272829 + 0.01827802843i\)
\(L(\frac12)\) \(\approx\) \(2.100272829 + 0.01827802843i\)
\(L(1)\) \(\approx\) \(1.658291724 - 0.07928547799i\)
\(L(1)\) \(\approx\) \(1.658291724 - 0.07928547799i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
good2 \( 1 + (0.957 - 0.289i)T \)
3 \( 1 + (-0.999 + 0.0183i)T \)
5 \( 1 + (0.606 + 0.794i)T \)
7 \( 1 + (0.993 - 0.110i)T \)
11 \( 1 + (0.137 + 0.990i)T \)
13 \( 1 + (-0.777 - 0.628i)T \)
17 \( 1 + (-0.896 + 0.443i)T \)
23 \( 1 + (0.484 + 0.875i)T \)
29 \( 1 + (0.280 + 0.959i)T \)
31 \( 1 + (0.451 - 0.892i)T \)
37 \( 1 + (0.789 - 0.614i)T \)
41 \( 1 + (0.663 - 0.748i)T \)
43 \( 1 + (-0.912 - 0.410i)T \)
47 \( 1 + (0.209 - 0.977i)T \)
53 \( 1 + (0.741 + 0.670i)T \)
59 \( 1 + (-0.979 - 0.200i)T \)
61 \( 1 + (-0.333 - 0.942i)T \)
67 \( 1 + (-0.861 + 0.507i)T \)
71 \( 1 + (-0.333 + 0.942i)T \)
73 \( 1 + (0.0642 - 0.997i)T \)
79 \( 1 + (0.100 + 0.994i)T \)
83 \( 1 + (-0.298 + 0.954i)T \)
89 \( 1 + (0.0642 + 0.997i)T \)
97 \( 1 + (-0.861 - 0.507i)T \)
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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

−24.45538192029849565552379021644, −24.053311512235458785613924500377, −23.020569340187637717980665239732, −21.98105691695189592670283670593, −21.44391794396285493820162678046, −20.840188555847725651239374518799, −19.63090279964892888515303023293, −18.211989499317156396871673046444, −17.30579136531470412276738789851, −16.70164936684836127153308140684, −15.96771065518555629000645098550, −14.797378260460993355279931406781, −13.80828282724800624343712264078, −13.06428799413361929215032372483, −11.97673827873396902250360058306, −11.47508594876372706001253989772, −10.45293087550918942311095261156, −8.99433553405540501356733885742, −7.88112621746893597926753835442, −6.61057904658060742992809571883, −5.87806468527119083086454651607, −4.70663704998799165942994444529, −4.56803281552522226278459028893, −2.52344352149735874711393835953, −1.28734741184071219958532520099, 1.52650603212076212753452992240, 2.44274027711193587754043713779, 4.04908092877398561406157286915, 4.98838868978023169687117444201, 5.72945171753165446838204574620, 6.86658712179805787758081516382, 7.480635165461501552066700842787, 9.61672144059900608452586899244, 10.517339313378499205852171503238, 11.0998940609686911893010172667, 12.02618337553605358207512738312, 12.91444981577540710613973681751, 13.873587131524790988259444961035, 15.065690896184757916528715842406, 15.25596821465990779901914050837, 16.90582039108460010531508883837, 17.61162898481546738802845307207, 18.32610545578032128981707822813, 19.61297206865038219405229030576, 20.59763056826098923992151875155, 21.621370706473469009235109138139, 22.01919008825776178966554883729, 22.91289479636001986056496955876, 23.56558554155342046767310792406, 24.56185128567829577704764468943

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