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Properties

Label 2-819-91.45-c1-0-16
Degree $2$
Conductor $819$
Sign $0.656 - 0.754i$
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  + (−1.74 + 1.74i)2-s − 4.08i·4-s + (0.638 − 2.38i)5-s + (−2.04 + 1.67i)7-s + (3.63 + 3.63i)8-s + (3.04 + 5.26i)10-s + (−1.17 + 4.40i)11-s + (1.54 − 3.25i)13-s + (0.640 − 6.49i)14-s − 4.49·16-s − 0.112·17-s + (−3.32 + 0.891i)19-s + (−9.72 − 2.60i)20-s + (−5.61 − 9.73i)22-s + 0.652i·23-s + ⋯
L(s)  = 1  + (−1.23 + 1.23i)2-s − 2.04i·4-s + (0.285 − 1.06i)5-s + (−0.773 + 0.634i)7-s + (1.28 + 1.28i)8-s + (0.962 + 1.66i)10-s + (−0.355 + 1.32i)11-s + (0.429 − 0.903i)13-s + (0.171 − 1.73i)14-s − 1.12·16-s − 0.0273·17-s + (−0.763 + 0.204i)19-s + (−2.17 − 0.582i)20-s + (−1.19 − 2.07i)22-s + 0.136i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.656 - 0.754i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.656 - 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.656 - 0.754i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.656 - 0.754i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.644512 + 0.293624i\)
\(L(\frac12)\) \(\approx\) \(0.644512 + 0.293624i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.04 - 1.67i)T \)
13 \( 1 + (-1.54 + 3.25i)T \)
good2 \( 1 + (1.74 - 1.74i)T - 2iT^{2} \)
5 \( 1 + (-0.638 + 2.38i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (1.17 - 4.40i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + 0.112T + 17T^{2} \)
19 \( 1 + (3.32 - 0.891i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 - 0.652iT - 23T^{2} \)
29 \( 1 + (-2.82 + 4.88i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5.34 + 1.43i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-1.09 - 1.09i)T + 37iT^{2} \)
41 \( 1 + (-10.6 + 2.86i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-6.08 + 3.51i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.72 - 1.53i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-2.41 + 4.17i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.79 - 2.79i)T - 59iT^{2} \)
61 \( 1 + (-13.2 - 7.66i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.07 + 1.62i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (3.56 + 0.955i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (0.651 + 2.43i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-6.11 - 10.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.34 - 3.34i)T + 83iT^{2} \)
89 \( 1 + (-6.14 + 6.14i)T - 89iT^{2} \)
97 \( 1 + (-1.04 + 3.89i)T + (-84.0 - 48.5i)T^{2} \)
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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

−9.931543048096819595248638678043, −9.334446126798089003766266060585, −8.625404409688108513917471517178, −7.952995940987072950055476569303, −7.04675764335054915908100275623, −6.03697237375542169828728255449, −5.50626121882066516390505342347, −4.38091379126872601883758727237, −2.35531029319798731964940196345, −0.78204962176980693930661874989, 0.863012266430644170760134734413, 2.48490068702843045200169707038, 3.15412789623734747218720552287, 4.12528878891882856797319944283, 6.11004947246253989887324847465, 6.79536150554143866433249948218, 7.81583837748366756649118446553, 8.756071322728807541075483478647, 9.393711330364854232089056437150, 10.37648374735394681544608607032

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