Properties

Label 2-819-91.6-c1-0-38
Degree $2$
Conductor $819$
Sign $-0.782 + 0.622i$
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  + (−0.0578 − 0.215i)2-s + (1.68 − 0.975i)4-s + (−1.08 − 1.08i)5-s + (−0.725 + 2.54i)7-s + (−0.624 − 0.624i)8-s + (−0.171 + 0.297i)10-s + (−4.17 + 1.11i)11-s + (−3.01 − 1.98i)13-s + (0.591 + 0.00954i)14-s + (1.85 − 3.20i)16-s + (−3.78 − 6.55i)17-s + (1.31 − 4.91i)19-s + (−2.89 − 0.776i)20-s + (0.482 + 0.836i)22-s + (1.85 + 1.07i)23-s + ⋯
L(s)  = 1  + (−0.0408 − 0.152i)2-s + (0.844 − 0.487i)4-s + (−0.486 − 0.486i)5-s + (−0.274 + 0.961i)7-s + (−0.220 − 0.220i)8-s + (−0.0543 + 0.0941i)10-s + (−1.25 + 0.337i)11-s + (−0.835 − 0.550i)13-s + (0.157 + 0.00255i)14-s + (0.462 − 0.801i)16-s + (−0.917 − 1.58i)17-s + (0.302 − 1.12i)19-s + (−0.647 − 0.173i)20-s + (0.102 + 0.178i)22-s + (0.387 + 0.223i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.782 + 0.622i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.782 + 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.782 + 0.622i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (370, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.782 + 0.622i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.295264 - 0.845797i\)
\(L(\frac12)\) \(\approx\) \(0.295264 - 0.845797i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.725 - 2.54i)T \)
13 \( 1 + (3.01 + 1.98i)T \)
good2 \( 1 + (0.0578 + 0.215i)T + (-1.73 + i)T^{2} \)
5 \( 1 + (1.08 + 1.08i)T + 5iT^{2} \)
11 \( 1 + (4.17 - 1.11i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (3.78 + 6.55i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.31 + 4.91i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-1.85 - 1.07i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.66 + 4.61i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.09 - 1.09i)T + 31iT^{2} \)
37 \( 1 + (2.82 - 0.757i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (1.37 - 0.369i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (6.58 - 3.80i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.26 - 5.26i)T - 47iT^{2} \)
53 \( 1 - 13.1T + 53T^{2} \)
59 \( 1 + (-8.73 - 2.34i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (10.1 - 5.88i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.28 + 4.78i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-5.25 - 1.40i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-8.04 + 8.04i)T - 73iT^{2} \)
79 \( 1 - 9.06T + 79T^{2} \)
83 \( 1 + (1.13 + 1.13i)T + 83iT^{2} \)
89 \( 1 + (-0.217 - 0.811i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (4.35 - 16.2i)T + (-84.0 - 48.5i)T^{2} \)
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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

−9.885894997404435870425868459506, −9.215420204095080764979242416144, −8.150026277421256406890127370619, −7.29951312584899111813935477217, −6.49995384805531622154840140873, −5.19639198176947695228197175270, −4.89210760896988983664535612566, −2.83610756797210915865482768787, −2.44655740443448579453339769707, −0.40104192854902978811136781373, 1.97364898452298833740996278050, 3.21904689918166397285275971911, 3.96290279280005729305649494355, 5.33650082049102067541567890965, 6.58785456063031077093947285496, 7.07662620335664486011870604495, 7.915445892086104216805957880339, 8.537085386038699645077362624035, 10.07490319726843094340110039089, 10.60007529454627740184269806797

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