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Properties

Label 2-819-91.45-c1-0-17
Degree $2$
Conductor $819$
Sign $0.356 - 0.934i$
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  + (−1.14 + 1.14i)2-s − 0.630i·4-s + (−0.395 + 1.47i)5-s + (0.0531 − 2.64i)7-s + (−1.57 − 1.57i)8-s + (−1.23 − 2.14i)10-s + (0.745 − 2.78i)11-s + (−2.94 + 2.08i)13-s + (2.97 + 3.09i)14-s + 4.86·16-s + 6.21·17-s + (2.23 − 0.598i)19-s + (0.930 + 0.249i)20-s + (2.33 + 4.04i)22-s + 5.62i·23-s + ⋯
L(s)  = 1  + (−0.811 + 0.811i)2-s − 0.315i·4-s + (−0.176 + 0.659i)5-s + (0.0200 − 0.999i)7-s + (−0.555 − 0.555i)8-s + (−0.391 − 0.678i)10-s + (0.224 − 0.838i)11-s + (−0.816 + 0.577i)13-s + (0.794 + 0.827i)14-s + 1.21·16-s + 1.50·17-s + (0.512 − 0.137i)19-s + (0.208 + 0.0557i)20-s + (0.498 + 0.862i)22-s + 1.17i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.356 - 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.356 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.356 - 0.934i$
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: \((2,\ 819,\ (\ :1/2),\ 0.356 - 0.934i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.767753 + 0.529068i\)
\(L(\frac12)\) \(\approx\) \(0.767753 + 0.529068i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0531 + 2.64i)T \)
13 \( 1 + (2.94 - 2.08i)T \)
good2 \( 1 + (1.14 - 1.14i)T - 2iT^{2} \)
5 \( 1 + (0.395 - 1.47i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-0.745 + 2.78i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 - 6.21T + 17T^{2} \)
19 \( 1 + (-2.23 + 0.598i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 - 5.62iT - 23T^{2} \)
29 \( 1 + (0.379 - 0.656i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-8.36 + 2.24i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (4.26 + 4.26i)T + 37iT^{2} \)
41 \( 1 + (-1.94 + 0.522i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (2.24 - 1.29i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.13 - 0.571i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (2.47 - 4.28i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.623 + 0.623i)T - 59iT^{2} \)
61 \( 1 + (-4.48 - 2.58i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-15.1 - 4.06i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-10.3 - 2.76i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-1.80 - 6.72i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (4.24 + 7.35i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.51 - 1.51i)T + 83iT^{2} \)
89 \( 1 + (-5.91 + 5.91i)T - 89iT^{2} \)
97 \( 1 + (0.933 - 3.48i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.02748654149624939242599862266, −9.631231672825134716265664608390, −8.528145464935249111338757880812, −7.61874698687076445746956156430, −7.23575966657788677858433314484, −6.40495432628386164324979316973, −5.33563458861054315027764396125, −3.83639559061187629047937697363, −3.07884405827783616819161541421, −0.936994840731067834838693482988, 0.882411219284159720046226623756, 2.19710634593649542569477246392, 3.15415607260284277431541189214, 4.84311420553865452745130822645, 5.42702578808035568049782310004, 6.64704522570732494917186617188, 8.058685134448591037279888185018, 8.404584656334756071256987725378, 9.540013369687148944690670610404, 9.855610730162522842240509992994

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