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Properties

Label 2-817-817.740-c0-0-0
Degree $2$
Conductor $817$
Sign $0.359 - 0.932i$
Analytic cond. $0.407736$
Root an. cond. $0.638542$
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  + (−0.222 + 0.974i)4-s + (1.03 + 0.702i)5-s + (−0.365 + 0.632i)7-s + (0.955 + 0.294i)9-s + (−0.367 − 1.61i)11-s + (−0.900 − 0.433i)16-s + (−1.63 + 1.11i)17-s + (0.955 − 0.294i)19-s + (−0.914 + 0.848i)20-s + (−0.109 + 0.101i)23-s + (0.202 + 0.516i)25-s + (−0.535 − 0.496i)28-s + (−0.820 + 0.395i)35-s + (−0.5 + 0.866i)36-s + 43-s + 1.65·44-s + ⋯
L(s)  = 1  + (−0.222 + 0.974i)4-s + (1.03 + 0.702i)5-s + (−0.365 + 0.632i)7-s + (0.955 + 0.294i)9-s + (−0.367 − 1.61i)11-s + (−0.900 − 0.433i)16-s + (−1.63 + 1.11i)17-s + (0.955 − 0.294i)19-s + (−0.914 + 0.848i)20-s + (−0.109 + 0.101i)23-s + (0.202 + 0.516i)25-s + (−0.535 − 0.496i)28-s + (−0.820 + 0.395i)35-s + (−0.5 + 0.866i)36-s + 43-s + 1.65·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 817 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.359 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 817 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.359 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(817\)    =    \(19 \cdot 43\)
Sign: $0.359 - 0.932i$
Analytic conductor: \(0.407736\)
Root analytic conductor: \(0.638542\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{817} (740, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 817,\ (\ :0),\ 0.359 - 0.932i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.066079236\)
\(L(\frac12)\) \(\approx\) \(1.066079236\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 + (-0.955 + 0.294i)T \)
43 \( 1 - T \)
good2 \( 1 + (0.222 - 0.974i)T^{2} \)
3 \( 1 + (-0.955 - 0.294i)T^{2} \)
5 \( 1 + (-1.03 - 0.702i)T + (0.365 + 0.930i)T^{2} \)
7 \( 1 + (0.365 - 0.632i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.367 + 1.61i)T + (-0.900 + 0.433i)T^{2} \)
13 \( 1 + (0.988 + 0.149i)T^{2} \)
17 \( 1 + (1.63 - 1.11i)T + (0.365 - 0.930i)T^{2} \)
23 \( 1 + (0.109 - 0.101i)T + (0.0747 - 0.997i)T^{2} \)
29 \( 1 + (-0.955 + 0.294i)T^{2} \)
31 \( 1 + (0.733 + 0.680i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.222 - 0.974i)T^{2} \)
47 \( 1 + (-0.400 + 1.75i)T + (-0.900 - 0.433i)T^{2} \)
53 \( 1 + (0.988 - 0.149i)T^{2} \)
59 \( 1 + (-0.623 - 0.781i)T^{2} \)
61 \( 1 + (0.722 + 1.84i)T + (-0.733 + 0.680i)T^{2} \)
67 \( 1 + (-0.826 + 0.563i)T^{2} \)
71 \( 1 + (-0.0747 - 0.997i)T^{2} \)
73 \( 1 + (0.134 - 1.79i)T + (-0.988 - 0.149i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.722 + 0.108i)T + (0.955 + 0.294i)T^{2} \)
89 \( 1 + (-0.955 - 0.294i)T^{2} \)
97 \( 1 + (0.900 - 0.433i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.67867920833248504559014954239, −9.684008866391478027830837537666, −8.896777232717242023586145560447, −8.186438358462624120081782917070, −7.07360491864160936535207860427, −6.31411880412400392176282945496, −5.45006874549775091730276010605, −4.11014107831587010186266261031, −3.05222841818151151495216623998, −2.16877590997109301863445663190, 1.25998903062232964204036724751, 2.30269238535723680247668384239, 4.36429972440208773449042717786, 4.78541726149068877027923609349, 5.83443184815195666248696918086, 6.84493759882176523548269588604, 7.42989202578668518818877240782, 9.187117373345137228983858128417, 9.424765337059102585864747869742, 10.09196331999986205958602422761

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