Properties

Label 2-819-117.2-c1-0-4
Degree $2$
Conductor $819$
Sign $-0.642 - 0.766i$
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  + (0.236 + 0.236i)2-s + (−1.70 + 0.276i)3-s − 1.88i·4-s + (−0.283 + 1.05i)5-s + (−0.468 − 0.338i)6-s + (−0.258 + 0.965i)7-s + (0.917 − 0.917i)8-s + (2.84 − 0.944i)9-s + (−0.316 + 0.182i)10-s + (−3.58 + 3.58i)11-s + (0.521 + 3.22i)12-s + (0.743 − 3.52i)13-s + (−0.289 + 0.166i)14-s + (0.192 − 1.88i)15-s − 3.34·16-s + (1.27 − 2.20i)17-s + ⋯
L(s)  = 1  + (0.166 + 0.166i)2-s + (−0.987 + 0.159i)3-s − 0.944i·4-s + (−0.126 + 0.473i)5-s + (−0.191 − 0.138i)6-s + (−0.0978 + 0.365i)7-s + (0.324 − 0.324i)8-s + (0.949 − 0.314i)9-s + (−0.100 + 0.0578i)10-s + (−1.08 + 1.08i)11-s + (0.150 + 0.932i)12-s + (0.206 − 0.978i)13-s + (−0.0772 + 0.0446i)14-s + (0.0497 − 0.487i)15-s − 0.835·16-s + (0.308 − 0.534i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.642 - 0.766i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (470, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.642 - 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.209844 + 0.449753i\)
\(L(\frac12)\) \(\approx\) \(0.209844 + 0.449753i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.70 - 0.276i)T \)
7 \( 1 + (0.258 - 0.965i)T \)
13 \( 1 + (-0.743 + 3.52i)T \)
good2 \( 1 + (-0.236 - 0.236i)T + 2iT^{2} \)
5 \( 1 + (0.283 - 1.05i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (3.58 - 3.58i)T - 11iT^{2} \)
17 \( 1 + (-1.27 + 2.20i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.108 + 0.404i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (3.29 - 5.70i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 5.65iT - 29T^{2} \)
31 \( 1 + (6.24 + 1.67i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (2.91 - 10.8i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (3.39 - 0.909i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (0.278 - 0.160i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.30 - 4.88i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 - 4.23iT - 53T^{2} \)
59 \( 1 + (3.16 - 3.16i)T - 59iT^{2} \)
61 \( 1 + (-4.21 - 7.30i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.137 - 0.514i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (14.8 - 3.98i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (10.8 + 10.8i)T + 73iT^{2} \)
79 \( 1 + (-2.89 + 5.01i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.76 + 0.471i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (-7.86 - 2.10i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (10.9 + 2.92i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.36687518897205278136717017271, −10.10976001769131847195569952438, −9.125292375990369139713541936219, −7.57957643751724271661210767171, −7.04931176726592295525827396822, −5.93876507920227003402696823861, −5.33041136913160188526745920117, −4.65644197310991744195568873040, −3.11904027278051411968450563542, −1.53319978280638124970006924258, 0.26193853905297514254874415644, 2.12003652166012966975494623256, 3.65263227362557640767035498223, 4.44009716247818568799115479849, 5.44316135241294761523550886406, 6.41502753179804007121703220830, 7.35407733648132969840215110725, 8.189622446212189338782874351058, 8.894673412243298020888623868300, 10.26578192163333489462567794564

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