Properties

Label 2-819-91.59-c1-0-5
Degree $2$
Conductor $819$
Sign $-0.860 - 0.508i$
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  + (0.745 + 0.745i)2-s − 0.887i·4-s + (−3.80 − 1.01i)5-s + (0.148 + 2.64i)7-s + (2.15 − 2.15i)8-s + (−2.07 − 3.59i)10-s + (0.913 + 0.244i)11-s + (−0.783 + 3.51i)13-s + (−1.85 + 2.08i)14-s + 1.43·16-s − 7.96·17-s + (0.451 + 1.68i)19-s + (−0.904 + 3.37i)20-s + (0.499 + 0.864i)22-s + 6.93i·23-s + ⋯
L(s)  = 1  + (0.527 + 0.527i)2-s − 0.443i·4-s + (−1.70 − 0.455i)5-s + (0.0562 + 0.998i)7-s + (0.761 − 0.761i)8-s + (−0.656 − 1.13i)10-s + (0.275 + 0.0738i)11-s + (−0.217 + 0.976i)13-s + (−0.496 + 0.556i)14-s + 0.359·16-s − 1.93·17-s + (0.103 + 0.386i)19-s + (−0.202 + 0.754i)20-s + (0.106 + 0.184i)22-s + 1.44i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.860 - 0.508i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.860 - 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.860 - 0.508i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (514, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.860 - 0.508i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.166292 + 0.608070i\)
\(L(\frac12)\) \(\approx\) \(0.166292 + 0.608070i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.148 - 2.64i)T \)
13 \( 1 + (0.783 - 3.51i)T \)
good2 \( 1 + (-0.745 - 0.745i)T + 2iT^{2} \)
5 \( 1 + (3.80 + 1.01i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-0.913 - 0.244i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + 7.96T + 17T^{2} \)
19 \( 1 + (-0.451 - 1.68i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 - 6.93iT - 23T^{2} \)
29 \( 1 + (1.71 - 2.97i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.17 + 4.37i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (6.88 - 6.88i)T - 37iT^{2} \)
41 \( 1 + (-0.117 - 0.437i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-0.0936 + 0.0540i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.19 - 4.45i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (0.747 - 1.29i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.42 - 3.42i)T + 59iT^{2} \)
61 \( 1 + (5.73 + 3.30i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.17 + 4.36i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-2.68 + 10.0i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-3.95 + 1.05i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-0.473 - 0.820i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8.26 + 8.26i)T - 83iT^{2} \)
89 \( 1 + (3.79 + 3.79i)T + 89iT^{2} \)
97 \( 1 + (11.5 + 3.09i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.87704834759178297242958427409, −9.415919730458555935128577586353, −8.918845198679949800820354450200, −7.902068689611796333369602035625, −7.05679709704209107937659312264, −6.27760040759310410244901709707, −5.07475277504116248347325256124, −4.47116294906616294973157881745, −3.57534741387174055306378727613, −1.76507942801691314658182993128, 0.25667236959355315848973044428, 2.50838965473405391006269052568, 3.57188044264323567267584418968, 4.15879738695381818821780322647, 4.89158669224558384040790351494, 6.79207189043476719793517932060, 7.24176391794540666961138649905, 8.158149118346315845233236240300, 8.724300531437974794297220171196, 10.39702877606266362166550399603

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