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Properties

Label 2-819-117.41-c1-0-27
Degree $2$
Conductor $819$
Sign $0.800 - 0.598i$
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  + (−0.897 + 0.240i)2-s + (−0.920 + 1.46i)3-s + (−0.984 + 0.568i)4-s + (−2.48 + 0.667i)5-s + (0.473 − 1.53i)6-s + (−0.707 − 0.707i)7-s + (2.06 − 2.06i)8-s + (−1.30 − 2.70i)9-s + (2.07 − 1.19i)10-s + (0.573 + 2.14i)11-s + (0.0723 − 1.96i)12-s + (−1.08 − 3.43i)13-s + (0.804 + 0.464i)14-s + (1.31 − 4.26i)15-s + (−0.218 + 0.377i)16-s + (0.469 − 0.813i)17-s + ⋯
L(s)  = 1  + (−0.634 + 0.170i)2-s + (−0.531 + 0.847i)3-s + (−0.492 + 0.284i)4-s + (−1.11 + 0.298i)5-s + (0.193 − 0.628i)6-s + (−0.267 − 0.267i)7-s + (0.728 − 0.728i)8-s + (−0.435 − 0.900i)9-s + (0.656 − 0.378i)10-s + (0.173 + 0.645i)11-s + (0.0208 − 0.567i)12-s + (−0.300 − 0.953i)13-s + (0.215 + 0.124i)14-s + (0.339 − 1.10i)15-s + (−0.0545 + 0.0944i)16-s + (0.113 − 0.197i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.800 - 0.598i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.800 - 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.800 - 0.598i$
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: \((2,\ 819,\ (\ :1/2),\ 0.800 - 0.598i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.394699 + 0.131265i\)
\(L(\frac12)\) \(\approx\) \(0.394699 + 0.131265i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.920 - 1.46i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
13 \( 1 + (1.08 + 3.43i)T \)
good2 \( 1 + (0.897 - 0.240i)T + (1.73 - i)T^{2} \)
5 \( 1 + (2.48 - 0.667i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-0.573 - 2.14i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-0.469 + 0.813i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.0928 + 0.346i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + 3.13T + 23T^{2} \)
29 \( 1 + (-2.72 - 1.57i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.12 + 7.93i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (1.50 - 5.60i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-2.37 - 2.37i)T + 41iT^{2} \)
43 \( 1 + 2.53iT - 43T^{2} \)
47 \( 1 + (-7.54 - 2.02i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 - 13.1iT - 53T^{2} \)
59 \( 1 + (1.98 + 0.531i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 - 7.34T + 61T^{2} \)
67 \( 1 + (0.0555 - 0.0555i)T - 67iT^{2} \)
71 \( 1 + (4.59 - 1.23i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (4.23 + 4.23i)T + 73iT^{2} \)
79 \( 1 + (-1.26 - 2.19i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.48 + 9.25i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (-12.5 - 3.36i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-12.1 + 12.1i)T - 97iT^{2} \)
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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

−10.18247155498528607705188555747, −9.602845485350187239889601171209, −8.686244593683712601597785381579, −7.74276474575942066668721058275, −7.20857770291167972151344985727, −5.94863206216927778945448681753, −4.67700377349837916352710667265, −4.06052123610487758709837935645, −3.15515143007660275125641689584, −0.50949716997600040858491615016, 0.67043622726578636293027447837, 2.04393274457548457301141304530, 3.78557621653878885217498014391, 4.83538088207222893481154173624, 5.79132307535415245198688359724, 6.84467812946274677690268419760, 7.75726521438395770766131203154, 8.467820518024570945686856659790, 9.077193180784305281426831406544, 10.22845020867932278614557004205

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