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Properties

Label 2-819-91.80-c1-0-23
Degree $2$
Conductor $819$
Sign $0.956 - 0.292i$
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  + (0.185 + 0.693i)2-s + (1.28 − 0.741i)4-s + (−0.295 + 1.10i)5-s + (−0.650 − 2.56i)7-s + (1.76 + 1.76i)8-s − 0.818·10-s + (1.26 + 1.26i)11-s + (3.60 + 0.189i)13-s + (1.65 − 0.928i)14-s + (0.584 − 1.01i)16-s + (−1.84 − 3.19i)17-s + (−0.439 − 0.439i)19-s + (0.437 + 1.63i)20-s + (−0.642 + 1.11i)22-s + (1.96 + 1.13i)23-s + ⋯
L(s)  = 1  + (0.131 + 0.490i)2-s + (0.642 − 0.370i)4-s + (−0.131 + 0.492i)5-s + (−0.245 − 0.969i)7-s + (0.625 + 0.625i)8-s − 0.258·10-s + (0.381 + 0.381i)11-s + (0.998 + 0.0525i)13-s + (0.443 − 0.248i)14-s + (0.146 − 0.253i)16-s + (−0.447 − 0.775i)17-s + (−0.100 − 0.100i)19-s + (0.0978 + 0.365i)20-s + (−0.136 + 0.237i)22-s + (0.408 + 0.235i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.292i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.956 - 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.956 - 0.292i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (262, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.956 - 0.292i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.04280 + 0.305540i\)
\(L(\frac12)\) \(\approx\) \(2.04280 + 0.305540i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.650 + 2.56i)T \)
13 \( 1 + (-3.60 - 0.189i)T \)
good2 \( 1 + (-0.185 - 0.693i)T + (-1.73 + i)T^{2} \)
5 \( 1 + (0.295 - 1.10i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-1.26 - 1.26i)T + 11iT^{2} \)
17 \( 1 + (1.84 + 3.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.439 + 0.439i)T + 19iT^{2} \)
23 \( 1 + (-1.96 - 1.13i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.0743 - 0.128i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.64 + 1.24i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (4.33 - 1.16i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-1.12 + 4.19i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-6.81 - 3.93i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.291 - 0.0779i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-6.19 + 10.7i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (13.7 + 3.68i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + 2.90iT - 61T^{2} \)
67 \( 1 + (8.53 - 8.53i)T - 67iT^{2} \)
71 \( 1 + (-3.67 - 13.6i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-3.05 - 11.3i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (0.537 + 0.930i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.90 - 3.90i)T + 83iT^{2} \)
89 \( 1 + (2.99 + 11.1i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (4.98 - 1.33i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.41824345987338908517026839716, −9.533654556292400021255110318948, −8.426209581053742463550542136534, −7.31191089081025592983362671880, −6.90644771063200251236386999829, −6.15893411487857918917485107328, −4.99598162112947306386433272679, −3.92978978795949942100573803507, −2.74393631078122979014911706925, −1.23268483825370493267866913744, 1.35631106987363287769652092047, 2.62292624973626927840732932163, 3.58921938445362018409953721576, 4.62185662977123058419121117591, 6.00755853014600819757522994992, 6.49628829580096932521558005471, 7.77325666471234218457147468652, 8.655098397872605683789966591956, 9.151128285253435061767791156019, 10.56705891424937839311253542181

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