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Properties

Label 2-817-817.531-c0-0-0
Degree $2$
Conductor $817$
Sign $0.561 + 0.827i$
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 + (0.0931 + 1.24i)5-s + (0.988 − 1.71i)7-s + (−0.733 − 0.680i)9-s + (−0.0332 + 0.145i)11-s + (−0.900 + 0.433i)16-s + (0.0546 − 0.728i)17-s + (−0.733 + 0.680i)19-s + (1.19 − 0.367i)20-s + (1.57 − 0.487i)23-s + (−0.548 + 0.0827i)25-s + (−1.88 − 0.582i)28-s + (2.22 + 1.07i)35-s + (−0.5 + 0.866i)36-s + 43-s + 0.149·44-s + ⋯
L(s)  = 1  + (−0.222 − 0.974i)4-s + (0.0931 + 1.24i)5-s + (0.988 − 1.71i)7-s + (−0.733 − 0.680i)9-s + (−0.0332 + 0.145i)11-s + (−0.900 + 0.433i)16-s + (0.0546 − 0.728i)17-s + (−0.733 + 0.680i)19-s + (1.19 − 0.367i)20-s + (1.57 − 0.487i)23-s + (−0.548 + 0.0827i)25-s + (−1.88 − 0.582i)28-s + (2.22 + 1.07i)35-s + (−0.5 + 0.866i)36-s + 43-s + 0.149·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 817 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.561 + 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 817 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.561 + 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(817\)    =    \(19 \cdot 43\)
Sign: $0.561 + 0.827i$
Analytic conductor: \(0.407736\)
Root analytic conductor: \(0.638542\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{817} (531, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 817,\ (\ :0),\ 0.561 + 0.827i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9691352381\)
\(L(\frac12)\) \(\approx\) \(0.9691352381\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 + (0.733 - 0.680i)T \)
43 \( 1 - T \)
good2 \( 1 + (0.222 + 0.974i)T^{2} \)
3 \( 1 + (0.733 + 0.680i)T^{2} \)
5 \( 1 + (-0.0931 - 1.24i)T + (-0.988 + 0.149i)T^{2} \)
7 \( 1 + (-0.988 + 1.71i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.0332 - 0.145i)T + (-0.900 - 0.433i)T^{2} \)
13 \( 1 + (-0.365 + 0.930i)T^{2} \)
17 \( 1 + (-0.0546 + 0.728i)T + (-0.988 - 0.149i)T^{2} \)
23 \( 1 + (-1.57 + 0.487i)T + (0.826 - 0.563i)T^{2} \)
29 \( 1 + (0.733 - 0.680i)T^{2} \)
31 \( 1 + (-0.955 - 0.294i)T^{2} \)
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 \( 1 + (-0.365 - 0.930i)T^{2} \)
59 \( 1 + (-0.623 + 0.781i)T^{2} \)
61 \( 1 + (0.722 - 0.108i)T + (0.955 - 0.294i)T^{2} \)
67 \( 1 + (-0.0747 + 0.997i)T^{2} \)
71 \( 1 + (-0.826 - 0.563i)T^{2} \)
73 \( 1 + (1.48 - 1.01i)T + (0.365 - 0.930i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.722 - 1.84i)T + (-0.733 - 0.680i)T^{2} \)
89 \( 1 + (0.733 + 0.680i)T^{2} \)
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

−10.54010152728688692217142859904, −9.693699503681568388460205466005, −8.691154689556512616272571825774, −7.50712747116519508695659227227, −6.86906918134381757196668333040, −6.07503491550190610470745963772, −4.89487247650206022151892315798, −4.00127865347870762904306307810, −2.73015948272330959776428375001, −1.11995656962691323834649655472, 1.94836381140126406687420499517, 2.96042946561219285416612142635, 4.55664581382038860756969482343, 5.11888382235018037564269467429, 5.90601774059150855234454049801, 7.47123690440304396531014204951, 8.414897596130060180734866176695, 8.742019875106287840895018392646, 9.142196175121118066452420188677, 10.84006026122347162800805212842

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