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Properties

Label 2-819-117.2-c1-0-12
Degree $2$
Conductor $819$
Sign $-0.0599 - 0.998i$
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  + (−0.404 − 0.404i)2-s + (0.200 + 1.72i)3-s − 1.67i·4-s + (0.289 − 1.08i)5-s + (0.614 − 0.776i)6-s + (−0.258 + 0.965i)7-s + (−1.48 + 1.48i)8-s + (−2.91 + 0.689i)9-s + (−0.554 + 0.320i)10-s + (−0.192 + 0.192i)11-s + (2.87 − 0.335i)12-s + (−2.47 + 2.62i)13-s + (0.495 − 0.285i)14-s + (1.91 + 0.281i)15-s − 2.14·16-s + (−0.531 + 0.921i)17-s + ⋯
L(s)  = 1  + (−0.285 − 0.285i)2-s + (0.115 + 0.993i)3-s − 0.836i·4-s + (0.129 − 0.483i)5-s + (0.250 − 0.317i)6-s + (−0.0978 + 0.365i)7-s + (−0.525 + 0.525i)8-s + (−0.973 + 0.229i)9-s + (−0.175 + 0.101i)10-s + (−0.0580 + 0.0580i)11-s + (0.830 − 0.0968i)12-s + (−0.686 + 0.726i)13-s + (0.132 − 0.0764i)14-s + (0.495 + 0.0727i)15-s − 0.536·16-s + (−0.129 + 0.223i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0599 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0599 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.0599 - 0.998i$
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: \((2,\ 819,\ (\ :1/2),\ -0.0599 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.609847 + 0.647596i\)
\(L(\frac12)\) \(\approx\) \(0.609847 + 0.647596i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.200 - 1.72i)T \)
7 \( 1 + (0.258 - 0.965i)T \)
13 \( 1 + (2.47 - 2.62i)T \)
good2 \( 1 + (0.404 + 0.404i)T + 2iT^{2} \)
5 \( 1 + (-0.289 + 1.08i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (0.192 - 0.192i)T - 11iT^{2} \)
17 \( 1 + (0.531 - 0.921i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.72 - 6.41i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (0.0829 - 0.143i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 6.09iT - 29T^{2} \)
31 \( 1 + (-1.78 - 0.477i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (0.977 - 3.64i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-7.82 + 2.09i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (6.27 - 3.62i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.25 + 8.42i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + 2.84iT - 53T^{2} \)
59 \( 1 + (-2.05 + 2.05i)T - 59iT^{2} \)
61 \( 1 + (-6.60 - 11.4i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.111 + 0.414i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (11.7 - 3.14i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-1.37 - 1.37i)T + 73iT^{2} \)
79 \( 1 + (6.49 - 11.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (16.8 - 4.52i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (10.5 + 2.83i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (8.88 + 2.37i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.07567268676352692147013983785, −9.872264858331747790346820351113, −8.904865486318456058560947574147, −8.410052256285394159670620325418, −6.91906263658366990173417005947, −5.73185109992255307145651094454, −5.18490119355927847520353377303, −4.22128700113660050213047269367, −2.88200386640852552269295926127, −1.59298722164121693007985474269, 0.46800489824007280955772441368, 2.52167402484393795896668055204, 3.11558437726163626394388610905, 4.59096404266035273982104720864, 5.96507080750669744630990507931, 6.88514973703700087837677879722, 7.37693671019273842329367734489, 8.100831413040131546850675723077, 8.973922980146725072595232435097, 9.846409607472965336468443964790

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