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Properties

Label 2-819-117.103-c1-0-17
Degree $2$
Conductor $819$
Sign $-0.734 - 0.678i$
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  + (−1.95 + 1.12i)2-s + (0.472 + 1.66i)3-s + (1.53 − 2.66i)4-s + (−2.16 − 1.24i)5-s + (−2.79 − 2.71i)6-s + (0.866 − 0.5i)7-s + 2.41i·8-s + (−2.55 + 1.57i)9-s + 5.63·10-s + (−0.514 + 0.296i)11-s + (5.16 + 1.30i)12-s + (2.92 − 2.10i)13-s + (−1.12 + 1.95i)14-s + (1.06 − 4.19i)15-s + (0.349 + 0.604i)16-s + 3.19·17-s + ⋯
L(s)  = 1  + (−1.37 + 0.796i)2-s + (0.272 + 0.962i)3-s + (0.768 − 1.33i)4-s + (−0.968 − 0.558i)5-s + (−1.14 − 1.10i)6-s + (0.327 − 0.188i)7-s + 0.855i·8-s + (−0.851 + 0.524i)9-s + 1.78·10-s + (−0.155 + 0.0895i)11-s + (1.49 + 0.376i)12-s + (0.812 − 0.582i)13-s + (−0.301 + 0.521i)14-s + (0.273 − 1.08i)15-s + (0.0872 + 0.151i)16-s + 0.774·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.734 - 0.678i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.734 - 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.734 - 0.678i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.734 - 0.678i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.217138 + 0.555496i\)
\(L(\frac12)\) \(\approx\) \(0.217138 + 0.555496i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.472 - 1.66i)T \)
7 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 + (-2.92 + 2.10i)T \)
good2 \( 1 + (1.95 - 1.12i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (2.16 + 1.24i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.514 - 0.296i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 - 3.19T + 17T^{2} \)
19 \( 1 - 0.273iT - 19T^{2} \)
23 \( 1 + (1.91 - 3.31i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.81 - 3.15i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.35 - 1.93i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 8.05iT - 37T^{2} \)
41 \( 1 + (-5.65 - 3.26i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.16 - 5.47i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.24 - 1.87i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 7.48T + 53T^{2} \)
59 \( 1 + (-4.03 - 2.32i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.666 + 1.15i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.17 - 3.56i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 2.32iT - 71T^{2} \)
73 \( 1 - 4.48iT - 73T^{2} \)
79 \( 1 + (7.09 + 12.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.84 + 1.64i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 15.4iT - 89T^{2} \)
97 \( 1 + (12.2 - 7.06i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.25361502197616471154848183605, −9.580264597952401951690941336080, −8.696206000730541187405359786085, −8.013621562924937354606341626886, −7.78372072827998728222056910047, −6.37051585201193879721862396715, −5.33471101944171578240041414934, −4.29259669625674687730591705678, −3.24941968206716507757931941855, −1.10608858694204948422154976585, 0.55838949843153826559938629561, 1.89861765940363569425967077478, 2.90499028692247016923590995904, 3.94222115917075417185554008786, 5.79640395098202159100979850580, 6.93356196673011696861131712468, 7.69981596803771147459975468774, 8.241939803434927292982379644957, 8.907445525811388370322722923620, 9.875004393805538740036990212054

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