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Properties

Label 2-819-91.76-c1-0-7
Degree $2$
Conductor $819$
Sign $-0.863 - 0.504i$
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  + (−0.0473 + 0.176i)2-s + (1.70 + 0.983i)4-s + (−2.80 + 2.80i)5-s + (−0.467 + 2.60i)7-s + (−0.513 + 0.513i)8-s + (−0.362 − 0.627i)10-s + (2.53 + 0.679i)11-s + (1.37 − 3.33i)13-s + (−0.437 − 0.205i)14-s + (1.90 + 3.29i)16-s + (1.43 − 2.48i)17-s + (0.759 + 2.83i)19-s + (−7.52 + 2.01i)20-s + (−0.240 + 0.415i)22-s + (−7.27 + 4.19i)23-s + ⋯
L(s)  = 1  + (−0.0334 + 0.124i)2-s + (0.851 + 0.491i)4-s + (−1.25 + 1.25i)5-s + (−0.176 + 0.984i)7-s + (−0.181 + 0.181i)8-s + (−0.114 − 0.198i)10-s + (0.764 + 0.204i)11-s + (0.380 − 0.924i)13-s + (−0.117 − 0.0550i)14-s + (0.475 + 0.822i)16-s + (0.347 − 0.601i)17-s + (0.174 + 0.650i)19-s + (−1.68 + 0.450i)20-s + (−0.0511 + 0.0886i)22-s + (−1.51 + 0.875i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.863 - 0.504i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.863 - 0.504i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.863 - 0.504i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (622, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.863 - 0.504i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.320617 + 1.18375i\)
\(L(\frac12)\) \(\approx\) \(0.320617 + 1.18375i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.467 - 2.60i)T \)
13 \( 1 + (-1.37 + 3.33i)T \)
good2 \( 1 + (0.0473 - 0.176i)T + (-1.73 - i)T^{2} \)
5 \( 1 + (2.80 - 2.80i)T - 5iT^{2} \)
11 \( 1 + (-2.53 - 0.679i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-1.43 + 2.48i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.759 - 2.83i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (7.27 - 4.19i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.66 + 2.87i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (6.75 - 6.75i)T - 31iT^{2} \)
37 \( 1 + (6.77 + 1.81i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-2.79 - 0.747i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (2.43 + 1.40i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.85 - 4.85i)T + 47iT^{2} \)
53 \( 1 - 5.43T + 53T^{2} \)
59 \( 1 + (-0.00666 + 0.00178i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-5.65 - 3.26i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.10 + 7.84i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-14.5 + 3.91i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-0.321 - 0.321i)T + 73iT^{2} \)
79 \( 1 - 0.280T + 79T^{2} \)
83 \( 1 + (-2.42 + 2.42i)T - 83iT^{2} \)
89 \( 1 + (0.0536 - 0.200i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-0.197 - 0.736i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.80139420975429089831849417018, −9.876367723544330894664429367832, −8.639228589673795113634262208762, −7.81390519852879270671307723234, −7.28612007200736139328339713175, −6.38444965303033805212108540443, −5.57569086034951932158671322799, −3.69564586226623462524973542606, −3.35519116400175820390504074246, −2.14963948925691643984797997787, 0.60313546887642574997421657650, 1.75313769355544881271483610191, 3.73838038598313303157814050399, 4.13492567300863933525014402898, 5.40922757615739015905645933256, 6.55882775604222463874006936422, 7.27905549503066079532725177727, 8.171523574045828625599778891919, 9.007500400863345938620709271932, 9.908420670598743908874170729683

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