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Properties

Label 2-819-117.41-c1-0-41
Degree $2$
Conductor $819$
Sign $0.854 + 0.518i$
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  + (−0.177 + 0.0474i)2-s + (−0.650 − 1.60i)3-s + (−1.70 + 0.983i)4-s + (1.66 − 0.446i)5-s + (0.191 + 0.253i)6-s + (−0.707 − 0.707i)7-s + (0.514 − 0.514i)8-s + (−2.15 + 2.08i)9-s + (−0.273 + 0.158i)10-s + (0.724 + 2.70i)11-s + (2.68 + 2.09i)12-s + (1.29 + 3.36i)13-s + (0.158 + 0.0916i)14-s + (−1.79 − 2.38i)15-s + (1.89 − 3.29i)16-s + (3.00 − 5.21i)17-s + ⋯
L(s)  = 1  + (−0.125 + 0.0335i)2-s + (−0.375 − 0.926i)3-s + (−0.851 + 0.491i)4-s + (0.744 − 0.199i)5-s + (0.0780 + 0.103i)6-s + (−0.267 − 0.267i)7-s + (0.181 − 0.181i)8-s + (−0.718 + 0.695i)9-s + (−0.0865 + 0.0499i)10-s + (0.218 + 0.814i)11-s + (0.775 + 0.604i)12-s + (0.360 + 0.932i)13-s + (0.0424 + 0.0244i)14-s + (−0.464 − 0.615i)15-s + (0.474 − 0.822i)16-s + (0.729 − 1.26i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.854 + 0.518i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.854 + 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.854 + 0.518i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (743, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.854 + 0.518i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11599 - 0.312152i\)
\(L(\frac12)\) \(\approx\) \(1.11599 - 0.312152i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.650 + 1.60i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
13 \( 1 + (-1.29 - 3.36i)T \)
good2 \( 1 + (0.177 - 0.0474i)T + (1.73 - i)T^{2} \)
5 \( 1 + (-1.66 + 0.446i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-0.724 - 2.70i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-3.00 + 5.21i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.241 - 0.899i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 - 5.41T + 23T^{2} \)
29 \( 1 + (-0.340 - 0.196i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.25 - 4.66i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (-2.81 + 10.5i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (5.13 + 5.13i)T + 41iT^{2} \)
43 \( 1 + 10.1iT - 43T^{2} \)
47 \( 1 + (-12.1 - 3.25i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 - 1.96iT - 53T^{2} \)
59 \( 1 + (-3.58 - 0.961i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 - 15.2T + 61T^{2} \)
67 \( 1 + (-6.17 + 6.17i)T - 67iT^{2} \)
71 \( 1 + (-0.488 + 0.130i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (4.86 + 4.86i)T + 73iT^{2} \)
79 \( 1 + (-5.62 - 9.73i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.96 - 11.0i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (14.3 + 3.84i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-9.08 + 9.08i)T - 97iT^{2} \)
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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

−9.946915608708860016564921910098, −9.245791011593006703438560924899, −8.590485554232565544613867994416, −7.22490016018113809756204236250, −7.10548368055614422596225263586, −5.67917348829213235330678207852, −4.98237794378488274056442215689, −3.75327237124884945436739282775, −2.29837381961440836610003131317, −0.924567461040402433998489359557, 0.984561362568658292275171665834, 2.98675025371467500459264623129, 3.96227215464788312736991012492, 5.12720677576767281100180793924, 5.83809130285909410479087722099, 6.32679587004677952117196886714, 8.211620591541174219952080310241, 8.719022612511536451009867472410, 9.801161976071019339871227204924, 10.04087516582181395140483850591

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