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Properties

Label 2-819-91.88-c1-0-4
Degree $2$
Conductor $819$
Sign $0.313 - 0.949i$
Analytic cond. $6.53974$
Root an. cond. $2.55729$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.89 − 1.09i)2-s + (1.39 + 2.41i)4-s + (1.5 − 0.866i)5-s + 2.64i·7-s − 1.73i·8-s − 3.79·10-s + 3.46i·11-s + (−1 − 3.46i)13-s + (2.89 − 5.01i)14-s + (0.895 − 1.55i)16-s + (0.5 + 0.866i)17-s + 5.29i·19-s + (4.18 + 2.41i)20-s + (3.79 − 6.56i)22-s + (−4.29 + 7.43i)23-s + ⋯
L(s)  = 1  + (−1.34 − 0.773i)2-s + (0.697 + 1.20i)4-s + (0.670 − 0.387i)5-s + 0.999i·7-s − 0.612i·8-s − 1.19·10-s + 1.04i·11-s + (−0.277 − 0.960i)13-s + (0.773 − 1.34i)14-s + (0.223 − 0.387i)16-s + (0.121 + 0.210i)17-s + 1.21i·19-s + (0.936 + 0.540i)20-s + (0.808 − 1.40i)22-s + (−0.894 + 1.54i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.313 - 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.313 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.313 - 0.949i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.313 - 0.949i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.421460 + 0.304729i\)
\(L(\frac12)\) \(\approx\) \(0.421460 + 0.304729i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 - 2.64iT \)
13 \( 1 + (1 + 3.46i)T \)
good2 \( 1 + (1.89 + 1.09i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (-1.5 + 0.866i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 - 3.46iT - 11T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 5.29iT - 19T^{2} \)
23 \( 1 + (4.29 - 7.43i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.5 + 6.06i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (5.29 + 3.05i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (6.08 + 3.51i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.08 + 1.77i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.29 - 3.96i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.708 - 0.409i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.08 - 5.33i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.70 + 2.14i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + 5.16T + 61T^{2} \)
67 \( 1 - 14.0iT - 67T^{2} \)
71 \( 1 + (-3.87 - 2.23i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-7.5 - 4.33i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.708 - 1.22i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 3.46iT - 83T^{2} \)
89 \( 1 + (-13.5 - 7.79i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-9.08 - 5.24i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.956545898925691316386889997171, −9.735358097091517308555654662478, −9.026758459352816145685980539706, −7.986284850293082089929937080247, −7.52370920527329137855127871430, −5.84171041080062284154538081379, −5.35886190258294341255449690248, −3.64103800607468146538650234045, −2.24944361885843514871610019549, −1.64009290085811136060289688496, 0.38782598260289334595382310090, 1.90481891046691835189821380383, 3.50884742091356147411451585379, 4.88858586531360405248305658172, 6.23891223801361135210750759004, 6.72987749437301157966493606533, 7.45823743044141798716143960105, 8.487162633409369012476332534853, 9.103772304582207063351584544819, 9.934821851482085211200596923985

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