Properties

Label 2-819-91.4-c1-0-12
Degree $2$
Conductor $819$
Sign $-0.937 - 0.348i$
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  + 1.37i·2-s + 0.0982·4-s + (0.697 − 0.402i)5-s + (0.0699 + 2.64i)7-s + 2.89i·8-s + (0.555 + 0.962i)10-s + (−4.56 + 2.63i)11-s + (−2.36 − 2.72i)13-s + (−3.64 + 0.0965i)14-s − 3.79·16-s − 0.560·17-s + (5.06 + 2.92i)19-s + (0.0685 − 0.0395i)20-s + (−3.63 − 6.29i)22-s − 1.60·23-s + ⋯
L(s)  = 1  + 0.975i·2-s + 0.0491·4-s + (0.312 − 0.180i)5-s + (0.0264 + 0.999i)7-s + 1.02i·8-s + (0.175 + 0.304i)10-s + (−1.37 + 0.794i)11-s + (−0.656 − 0.754i)13-s + (−0.974 + 0.0257i)14-s − 0.948·16-s − 0.135·17-s + (1.16 + 0.670i)19-s + (0.0153 − 0.00884i)20-s + (−0.774 − 1.34i)22-s − 0.334·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.937 - 0.348i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.937 - 0.348i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.937 - 0.348i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (550, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.937 - 0.348i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.252860 + 1.40722i\)
\(L(\frac12)\) \(\approx\) \(0.252860 + 1.40722i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0699 - 2.64i)T \)
13 \( 1 + (2.36 + 2.72i)T \)
good2 \( 1 - 1.37iT - 2T^{2} \)
5 \( 1 + (-0.697 + 0.402i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (4.56 - 2.63i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + 0.560T + 17T^{2} \)
19 \( 1 + (-5.06 - 2.92i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 1.60T + 23T^{2} \)
29 \( 1 + (-1.14 + 1.97i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.01 - 1.73i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 1.24iT - 37T^{2} \)
41 \( 1 + (0.803 + 0.463i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.22 - 3.85i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.32 + 1.92i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.72 + 4.72i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 10.9iT - 59T^{2} \)
61 \( 1 + (3.65 - 6.32i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.36 + 3.67i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-8.06 + 4.65i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-4.33 - 2.50i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.68 + 9.84i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 5.81iT - 83T^{2} \)
89 \( 1 + 5.00iT - 89T^{2} \)
97 \( 1 + (-9.22 + 5.32i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.37167457555197698694103058998, −9.745990971639688071901785331748, −8.687761542873773675645635361965, −7.79166170021658643513769807664, −7.39817957994397067459109118324, −6.10720370498388741251707275427, −5.43312423117105453012434144629, −4.89459793647316312627604413028, −2.94494743388972853970230904706, −2.06848503092773324202582424983, 0.66073079644487281432416975410, 2.20542738563265175864166555907, 3.06515776834280678016909051824, 4.15532822465887124993726475057, 5.25288132747272609209537016686, 6.49130688523390662890622775060, 7.27577540990975581260286754602, 8.105736098069650251529794648467, 9.461985615841985327806773316918, 10.03217599010286182468897967636

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