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Properties

Label 2-819-91.47-c1-0-12
Degree $2$
Conductor $819$
Sign $-0.218 - 0.975i$
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  + (−2.38 + 0.639i)2-s + (3.56 − 2.05i)4-s + (0.423 + 1.58i)5-s + (−0.716 + 2.54i)7-s + (−3.69 + 3.69i)8-s + (−2.02 − 3.50i)10-s + (5.55 + 1.48i)11-s + (3.57 + 0.473i)13-s + (0.0809 − 6.54i)14-s + (2.34 − 4.06i)16-s + (0.991 + 1.71i)17-s + (−0.246 − 0.918i)19-s + (4.76 + 4.76i)20-s − 14.2·22-s + (−3.06 − 1.77i)23-s + ⋯
L(s)  = 1  + (−1.68 + 0.452i)2-s + (1.78 − 1.02i)4-s + (0.189 + 0.707i)5-s + (−0.270 + 0.962i)7-s + (−1.30 + 1.30i)8-s + (−0.640 − 1.10i)10-s + (1.67 + 0.448i)11-s + (0.991 + 0.131i)13-s + (0.0216 − 1.74i)14-s + (0.587 − 1.01i)16-s + (0.240 + 0.416i)17-s + (−0.0564 − 0.210i)19-s + (1.06 + 1.06i)20-s − 3.03·22-s + (−0.639 − 0.369i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.218 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.218 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.218 - 0.975i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (775, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.218 - 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.491241 + 0.613580i\)
\(L(\frac12)\) \(\approx\) \(0.491241 + 0.613580i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.716 - 2.54i)T \)
13 \( 1 + (-3.57 - 0.473i)T \)
good2 \( 1 + (2.38 - 0.639i)T + (1.73 - i)T^{2} \)
5 \( 1 + (-0.423 - 1.58i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (-5.55 - 1.48i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-0.991 - 1.71i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.246 + 0.918i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (3.06 + 1.77i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 2.83T + 29T^{2} \)
31 \( 1 + (-4.33 - 1.16i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-1.00 - 3.73i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-4.02 + 4.02i)T - 41iT^{2} \)
43 \( 1 + 5.30iT - 43T^{2} \)
47 \( 1 + (0.448 - 0.120i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-6.31 - 10.9i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.10 + 11.5i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (4.38 + 2.52i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.57 - 5.87i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-4.84 - 4.84i)T + 71iT^{2} \)
73 \( 1 + (1.13 - 4.24i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (3.08 - 5.33i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (11.5 - 11.5i)T - 83iT^{2} \)
89 \( 1 + (-3.51 + 0.941i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (7.09 - 7.09i)T - 97iT^{2} \)
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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

−10.19253797217879024715670079404, −9.470205319441678538765546013859, −8.821253767291659793250019480685, −8.214283408518404028380680663617, −6.96444599814794227147985629124, −6.48655113701785462022898513572, −5.81008934776586084959746945316, −3.95019235654956520336338038760, −2.48214364922223208725283503551, −1.34149251585409768574834920195, 0.821349021762810953988791885116, 1.53239903170756864819018384460, 3.27815334160006934575336961168, 4.24599390102339657999894730887, 5.97118214936863723446868741092, 6.80861046685691800722605260352, 7.72370429789922461972034278747, 8.556873196419705124538183492822, 9.221345848419713827816702048781, 9.779418087305595755273964842322

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