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Properties

Label 2-819-91.41-c1-0-8
Degree $2$
Conductor $819$
Sign $-0.402 - 0.915i$
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  + (1.38 + 0.369i)2-s + (0.0377 + 0.0217i)4-s + (0.512 + 0.512i)5-s + (−1.54 + 2.15i)7-s + (−1.97 − 1.97i)8-s + (0.517 + 0.896i)10-s + (−1.38 + 5.18i)11-s + (0.0545 + 3.60i)13-s + (−2.92 + 2.39i)14-s + (−2.04 − 3.53i)16-s + (−1.31 + 2.28i)17-s + (5.26 − 1.41i)19-s + (0.00817 + 0.0304i)20-s + (−3.83 + 6.64i)22-s + (−5.51 + 3.18i)23-s + ⋯
L(s)  = 1  + (0.976 + 0.261i)2-s + (0.0188 + 0.0108i)4-s + (0.229 + 0.229i)5-s + (−0.582 + 0.812i)7-s + (−0.699 − 0.699i)8-s + (0.163 + 0.283i)10-s + (−0.419 + 1.56i)11-s + (0.0151 + 0.999i)13-s + (−0.781 + 0.641i)14-s + (−0.510 − 0.884i)16-s + (−0.319 + 0.553i)17-s + (1.20 − 0.323i)19-s + (0.00182 + 0.00681i)20-s + (−0.818 + 1.41i)22-s + (−1.14 + 0.663i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.402 - 0.915i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.402 - 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.402 - 0.915i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (496, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.402 - 0.915i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.935193 + 1.43341i\)
\(L(\frac12)\) \(\approx\) \(0.935193 + 1.43341i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.54 - 2.15i)T \)
13 \( 1 + (-0.0545 - 3.60i)T \)
good2 \( 1 + (-1.38 - 0.369i)T + (1.73 + i)T^{2} \)
5 \( 1 + (-0.512 - 0.512i)T + 5iT^{2} \)
11 \( 1 + (1.38 - 5.18i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (1.31 - 2.28i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.26 + 1.41i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (5.51 - 3.18i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.300 + 0.520i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-6.22 - 6.22i)T + 31iT^{2} \)
37 \( 1 + (-0.172 + 0.644i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (2.11 - 7.88i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-4.10 - 2.36i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.25 + 4.25i)T - 47iT^{2} \)
53 \( 1 + 0.282T + 53T^{2} \)
59 \( 1 + (1.21 + 4.54i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (13.0 + 7.55i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.48 - 1.20i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (4.11 + 15.3i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (3.04 - 3.04i)T - 73iT^{2} \)
79 \( 1 - 4.77T + 79T^{2} \)
83 \( 1 + (-2.42 - 2.42i)T + 83iT^{2} \)
89 \( 1 + (5.75 + 1.54i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-15.7 + 4.21i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.26050716432833962006677388977, −9.668914470353391173665437867563, −9.043710636559129921228038380019, −7.74532163915269815610500985263, −6.65572724427203481122704446335, −6.16275592608859821395334939021, −5.06458542129921660235455621483, −4.39589552125912214521692983614, −3.19681194661837383058667116679, −2.04412388339919428228494905275, 0.60340341728026571284332565268, 2.77407498002861744477959822347, 3.45760899030429913472203075914, 4.42303036082037108715625515129, 5.63116403088754933695090674753, 5.95417840506552736077068739657, 7.41339836398103744116597562328, 8.244440312605787885272105853400, 9.144766477912750792718519475427, 10.11591262444494602589139252824

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