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Properties

Label 1-19e2-361.20-r0-0-0
Degree $1$
Conductor $361$
Sign $-0.433 - 0.901i$
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  + (−0.401 − 0.915i)2-s + (0.789 + 0.614i)3-s + (−0.677 + 0.735i)4-s + (−0.0825 − 0.996i)5-s + (0.245 − 0.969i)6-s + (−0.677 − 0.735i)7-s + (0.945 + 0.324i)8-s + (0.245 + 0.969i)9-s + (−0.879 + 0.475i)10-s + (0.245 − 0.969i)11-s + (−0.986 + 0.164i)12-s + (0.789 + 0.614i)13-s + (−0.401 + 0.915i)14-s + (0.546 − 0.837i)15-s + (−0.0825 − 0.996i)16-s + (−0.677 − 0.735i)17-s + ⋯
L(s)  = 1  + (−0.401 − 0.915i)2-s + (0.789 + 0.614i)3-s + (−0.677 + 0.735i)4-s + (−0.0825 − 0.996i)5-s + (0.245 − 0.969i)6-s + (−0.677 − 0.735i)7-s + (0.945 + 0.324i)8-s + (0.245 + 0.969i)9-s + (−0.879 + 0.475i)10-s + (0.245 − 0.969i)11-s + (−0.986 + 0.164i)12-s + (0.789 + 0.614i)13-s + (−0.401 + 0.915i)14-s + (0.546 − 0.837i)15-s + (−0.0825 − 0.996i)16-s + (−0.677 − 0.735i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.433 - 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.433 - 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(361\)    =    \(19^{2}\)
Sign: $-0.433 - 0.901i$
Analytic conductor: \(1.67647\)
Root analytic conductor: \(1.67647\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{361} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 361,\ (0:\ ),\ -0.433 - 0.901i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5812363746 - 0.9243944992i\)
\(L(\frac12)\) \(\approx\) \(0.5812363746 - 0.9243944992i\)
\(L(1)\) \(\approx\) \(0.8359406052 - 0.5029758303i\)
\(L(1)\) \(\approx\) \(0.8359406052 - 0.5029758303i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
good2 \( 1 + (-0.401 - 0.915i)T \)
3 \( 1 + (0.789 + 0.614i)T \)
5 \( 1 + (-0.0825 - 0.996i)T \)
7 \( 1 + (-0.677 - 0.735i)T \)
11 \( 1 + (0.245 - 0.969i)T \)
13 \( 1 + (0.789 + 0.614i)T \)
17 \( 1 + (-0.677 - 0.735i)T \)
23 \( 1 + (0.789 - 0.614i)T \)
29 \( 1 + (-0.677 - 0.735i)T \)
31 \( 1 + (-0.401 + 0.915i)T \)
37 \( 1 + (0.245 - 0.969i)T \)
41 \( 1 + (0.546 - 0.837i)T \)
43 \( 1 + (-0.879 - 0.475i)T \)
47 \( 1 + (0.245 - 0.969i)T \)
53 \( 1 + (0.245 - 0.969i)T \)
59 \( 1 + (0.546 - 0.837i)T \)
61 \( 1 + (0.945 - 0.324i)T \)
67 \( 1 + (0.945 + 0.324i)T \)
71 \( 1 + (0.945 + 0.324i)T \)
73 \( 1 + (-0.677 - 0.735i)T \)
79 \( 1 + (-0.879 - 0.475i)T \)
83 \( 1 + (-0.0825 + 0.996i)T \)
89 \( 1 + (-0.677 + 0.735i)T \)
97 \( 1 + (0.945 - 0.324i)T \)
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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

−25.406216108131809158073026413909, −24.34811003180242026770605614159, −23.31614088047468990903636478120, −22.682810479864598087538653731442, −21.74815504661033509872646161048, −20.20661359312665098801328702420, −19.466168144618844022588977598623, −18.60758996759599984816559415148, −18.15011723079290046447151929415, −17.22496997574637488722776494694, −15.724217059710246413168751707343, −15.10962738737808816450360916960, −14.64537987956718051161668720485, −13.34603118876796599331572277096, −12.81737552184562870359275734665, −11.278820714440736914422363312967, −10.00403720577363247540525198990, −9.24134207236543461226056817107, −8.2880891365394111676363312690, −7.31174591570959810571983548461, −6.577457049140179597664488387904, −5.79318062385873894324990313934, −4.02805050287175431509293199021, −2.885352952823179671334738219236, −1.60539481321680052663378274740, 0.730413561072604219275783113005, 2.15615295596239814750166633378, 3.53966572504443498533763768074, 4.03467945118305795244462109828, 5.189315156545889963055477718298, 7.02235692342076341849045638780, 8.33646669315624638639546810219, 8.94305851195862607563346057066, 9.59938281964450776271800165601, 10.72208118262641171271298522519, 11.48672224647668785114134351458, 12.89796256758750811573278579447, 13.4626194160296630345413807437, 14.18864049474203790289871714888, 15.9411587967331915825373899185, 16.38285825987371291462079227571, 17.19219654275307926715817017640, 18.65995313362936214001305016803, 19.39623521577379694295299787426, 20.1176604238523284952210642, 20.76736035382148131945543308328, 21.44803482345940535751371600944, 22.41572426571982556363144387843, 23.418448275603538102119669104454, 24.65779380791033218594392036724

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