Properties

Label 2-819-91.41-c1-0-21
Degree $2$
Conductor $819$
Sign $0.259 - 0.965i$
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  + (2.11 + 0.567i)2-s + (2.43 + 1.40i)4-s + (3.00 + 3.00i)5-s + (−2.60 + 0.481i)7-s + (1.26 + 1.26i)8-s + (4.65 + 8.06i)10-s + (0.698 − 2.60i)11-s + (0.373 + 3.58i)13-s + (−5.78 − 0.455i)14-s + (−0.857 − 1.48i)16-s + (−0.599 + 1.03i)17-s + (1.89 − 0.507i)19-s + (3.09 + 11.5i)20-s + (2.95 − 5.12i)22-s + (4.65 − 2.68i)23-s + ⋯
L(s)  = 1  + (1.49 + 0.401i)2-s + (1.21 + 0.703i)4-s + (1.34 + 1.34i)5-s + (−0.983 + 0.182i)7-s + (0.445 + 0.445i)8-s + (1.47 + 2.55i)10-s + (0.210 − 0.785i)11-s + (0.103 + 0.994i)13-s + (−1.54 − 0.121i)14-s + (−0.214 − 0.371i)16-s + (−0.145 + 0.251i)17-s + (0.434 − 0.116i)19-s + (0.691 + 2.57i)20-s + (0.630 − 1.09i)22-s + (0.970 − 0.560i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.259 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.259 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.259 - 0.965i$
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: \((2,\ 819,\ (\ :1/2),\ 0.259 - 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.06616 + 2.35149i\)
\(L(\frac12)\) \(\approx\) \(3.06616 + 2.35149i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.60 - 0.481i)T \)
13 \( 1 + (-0.373 - 3.58i)T \)
good2 \( 1 + (-2.11 - 0.567i)T + (1.73 + i)T^{2} \)
5 \( 1 + (-3.00 - 3.00i)T + 5iT^{2} \)
11 \( 1 + (-0.698 + 2.60i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (0.599 - 1.03i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.89 + 0.507i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-4.65 + 2.68i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.47 + 2.56i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.36 + 3.36i)T + 31iT^{2} \)
37 \( 1 + (1.03 - 3.87i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-1.42 + 5.32i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (9.78 + 5.64i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.97 + 2.97i)T - 47iT^{2} \)
53 \( 1 - 11.0T + 53T^{2} \)
59 \( 1 + (-3.14 - 11.7i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (4.55 + 2.63i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.15 - 1.11i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (0.800 + 2.98i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-4.78 + 4.78i)T - 73iT^{2} \)
79 \( 1 + 1.16T + 79T^{2} \)
83 \( 1 + (-3.24 - 3.24i)T + 83iT^{2} \)
89 \( 1 + (4.80 + 1.28i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (14.7 - 3.95i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.52843489610986978123397955004, −9.626743716363486318458675705476, −8.914271147540302068368334307495, −7.01734300979522847889759679993, −6.78144119122158585142079263577, −5.95222750410285988741150544166, −5.42091911842234235394933702339, −3.90873322373711871281185210110, −3.10970999440096201989146975787, −2.26318200108163526256549332782, 1.35379734487589164228753890301, 2.60311106421771857603736777723, 3.65887122282457717844733954746, 4.87204130774972665624560446112, 5.35357239377649177033166830565, 6.13182117927319971897145764947, 7.07698643382756098924351636690, 8.588721804441166734787364688766, 9.459509049691466369619614066110, 10.00581859077209774950981941348

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