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Properties

Label 2-819-91.74-c1-0-19
Degree $2$
Conductor $819$
Sign $0.995 + 0.0996i$
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  − 1.09·2-s − 0.803·4-s + (0.179 + 0.310i)5-s + (2.59 − 0.494i)7-s + 3.06·8-s + (−0.196 − 0.339i)10-s + (−0.697 − 1.20i)11-s + (3.19 + 1.66i)13-s + (−2.84 + 0.540i)14-s − 1.74·16-s − 6.02·17-s + (−0.0503 + 0.0871i)19-s + (−0.144 − 0.249i)20-s + (0.762 + 1.32i)22-s + 2.30·23-s + ⋯
L(s)  = 1  − 0.773·2-s − 0.401·4-s + (0.0802 + 0.139i)5-s + (0.982 − 0.186i)7-s + 1.08·8-s + (−0.0620 − 0.107i)10-s + (−0.210 − 0.364i)11-s + (0.887 + 0.461i)13-s + (−0.759 + 0.144i)14-s − 0.436·16-s − 1.46·17-s + (−0.0115 + 0.0199i)19-s + (−0.0322 − 0.0558i)20-s + (0.162 + 0.281i)22-s + 0.481·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.995 + 0.0996i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (802, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.995 + 0.0996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01855 - 0.0508810i\)
\(L(\frac12)\) \(\approx\) \(1.01855 - 0.0508810i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.59 + 0.494i)T \)
13 \( 1 + (-3.19 - 1.66i)T \)
good2 \( 1 + 1.09T + 2T^{2} \)
5 \( 1 + (-0.179 - 0.310i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.697 + 1.20i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + 6.02T + 17T^{2} \)
19 \( 1 + (0.0503 - 0.0871i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 2.30T + 23T^{2} \)
29 \( 1 + (4.04 - 7.00i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.94 + 5.10i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 5.28T + 37T^{2} \)
41 \( 1 + (-5.97 + 10.3i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.43 - 9.41i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.67 - 8.09i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.20 + 9.02i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 9.03T + 59T^{2} \)
61 \( 1 + (0.812 - 1.40i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.12 - 5.40i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.36 + 4.08i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (1.63 - 2.82i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.65 + 6.32i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 8.23T + 83T^{2} \)
89 \( 1 - 11.8T + 89T^{2} \)
97 \( 1 + (3.37 + 5.83i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.25330139867292543727437613875, −9.054624022905282797453215785724, −8.747701265079542792188646062374, −7.85897283676401215424551385791, −7.01607114240055261816622471822, −5.86983746228323494564595671260, −4.69044043889343476587013929717, −4.02523981735844145946949994462, −2.27819843109078879915636677048, −0.947595696948868870233630835117, 1.01531164636597339154416896920, 2.29878035210542005554528473894, 4.04614471370936541962968246676, 4.82092524919249583556363981548, 5.78402889683897263660959382319, 7.10387532822212868157198674370, 7.908407731825636063070426529284, 8.705843602976677174477473217443, 9.141898249871873696023520962124, 10.22954994303221863238044167504

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