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Properties

Label 1-373-373.78-r1-0-0
Degree $1$
Conductor $373$
Sign $0.837 + 0.545i$
Analytic cond. $40.0844$
Root an. cond. $40.0844$
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  + (0.747 − 0.664i)2-s + (−0.997 + 0.0675i)3-s + (0.117 − 0.993i)4-s + (−0.975 + 0.217i)5-s + (−0.701 + 0.713i)6-s + (0.758 + 0.651i)7-s + (−0.571 − 0.820i)8-s + (0.990 − 0.134i)9-s + (−0.585 + 0.810i)10-s + (−0.425 − 0.905i)11-s + (−0.0506 + 0.998i)12-s + (−0.820 − 0.571i)13-s + (0.999 − 0.0168i)14-s + (0.959 − 0.283i)15-s + (−0.972 − 0.234i)16-s + (−0.918 + 0.394i)17-s + ⋯
L(s)  = 1  + (0.747 − 0.664i)2-s + (−0.997 + 0.0675i)3-s + (0.117 − 0.993i)4-s + (−0.975 + 0.217i)5-s + (−0.701 + 0.713i)6-s + (0.758 + 0.651i)7-s + (−0.571 − 0.820i)8-s + (0.990 − 0.134i)9-s + (−0.585 + 0.810i)10-s + (−0.425 − 0.905i)11-s + (−0.0506 + 0.998i)12-s + (−0.820 − 0.571i)13-s + (0.999 − 0.0168i)14-s + (0.959 − 0.283i)15-s + (−0.972 − 0.234i)16-s + (−0.918 + 0.394i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.837 + 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.837 + 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $0.837 + 0.545i$
Analytic conductor: \(40.0844\)
Root analytic conductor: \(40.0844\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (78, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (1:\ ),\ 0.837 + 0.545i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8619506627 + 0.2560171998i\)
\(L(\frac12)\) \(\approx\) \(0.8619506627 + 0.2560171998i\)
\(L(1)\) \(\approx\) \(0.8420394235 - 0.2798692278i\)
\(L(1)\) \(\approx\) \(0.8420394235 - 0.2798692278i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (0.747 - 0.664i)T \)
3 \( 1 + (-0.997 + 0.0675i)T \)
5 \( 1 + (-0.975 + 0.217i)T \)
7 \( 1 + (0.758 + 0.651i)T \)
11 \( 1 + (-0.425 - 0.905i)T \)
13 \( 1 + (-0.820 - 0.571i)T \)
17 \( 1 + (-0.918 + 0.394i)T \)
19 \( 1 + (0.724 - 0.688i)T \)
23 \( 1 + (-0.897 + 0.440i)T \)
29 \( 1 + (0.470 + 0.882i)T \)
31 \( 1 + (0.0506 + 0.998i)T \)
37 \( 1 + (-0.638 - 0.769i)T \)
41 \( 1 + (-0.250 + 0.968i)T \)
43 \( 1 + (0.993 + 0.117i)T \)
47 \( 1 + (0.514 - 0.857i)T \)
53 \( 1 + (-0.134 + 0.990i)T \)
59 \( 1 + (0.999 + 0.0337i)T \)
61 \( 1 + (0.0675 + 0.997i)T \)
67 \( 1 + (0.299 - 0.954i)T \)
71 \( 1 + (0.801 - 0.598i)T \)
73 \( 1 + (-0.378 - 0.925i)T \)
79 \( 1 + (0.810 + 0.585i)T \)
83 \( 1 + (0.990 + 0.134i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (0.394 + 0.918i)T \)
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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

−24.22224092629263585750019906611, −23.499905513991025310359317992505, −22.680606337809338905054286415316, −22.174012426677681654226427266130, −20.8033987730242107984223882514, −20.34183029018349144843289243284, −18.865802469478545015194173171322, −17.6984361273188462289201765359, −17.19777681575947019627934267233, −16.173722725251617048255413837950, −15.58697639256936594081966963156, −14.59389564222635664328252783873, −13.568488026935274139253234785628, −12.4224067036040879863437171256, −11.87699033373222127769831990505, −11.11263358824711457361664245767, −9.83799745086115063103305875419, −8.16791615548036304290510994993, −7.41566459932189178093809010503, −6.78620105839572107940448336886, −5.3738058191896824320825692532, −4.50795503277773557568002086772, −4.08162817577648922642320343500, −2.13448712420424628831098866348, −0.277761846365871892400776580579, 0.927591831739932691681727971989, 2.466497671201278786323667060128, 3.67415366607020818150633587932, 4.84299750016632866259073878202, 5.3899440376646296515359356244, 6.56836771702589779706707384141, 7.719317779900590090668000064243, 9.045948916031059828985456535080, 10.551039221949770425233724415043, 10.98414898548090039317162721659, 11.922153363589659474672469914, 12.35921956254708376298750485297, 13.56517288944377197411729642053, 14.7422972392363925991791600308, 15.58171700806943814780842097845, 16.10675405260772211888677132788, 17.74121021086837702608335847496, 18.2862755085421526380317667957, 19.36185017884012045756404706104, 20.06729294268086943414231670882, 21.35391498379389574539953232862, 21.935217638395036873558467587944, 22.532286764482222501055266621034, 23.6656694669599720081597381115, 24.06360309365749260758272410935

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