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Properties

Label 2-819-117.2-c1-0-31
Degree $2$
Conductor $819$
Sign $0.408 + 0.912i$
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  + (−0.416 − 0.416i)2-s + (−1.43 − 0.970i)3-s − 1.65i·4-s + (−0.281 + 1.04i)5-s + (0.193 + 1.00i)6-s + (−0.258 + 0.965i)7-s + (−1.52 + 1.52i)8-s + (1.11 + 2.78i)9-s + (0.554 − 0.320i)10-s + (2.19 − 2.19i)11-s + (−1.60 + 2.37i)12-s + (3.08 + 1.87i)13-s + (0.510 − 0.294i)14-s + (1.42 − 1.23i)15-s − 2.03·16-s + (1.71 − 2.96i)17-s + ⋯
L(s)  = 1  + (−0.294 − 0.294i)2-s + (−0.828 − 0.560i)3-s − 0.826i·4-s + (−0.125 + 0.469i)5-s + (0.0790 + 0.408i)6-s + (−0.0978 + 0.365i)7-s + (−0.537 + 0.537i)8-s + (0.372 + 0.927i)9-s + (0.175 − 0.101i)10-s + (0.662 − 0.662i)11-s + (−0.462 + 0.684i)12-s + (0.854 + 0.518i)13-s + (0.136 − 0.0787i)14-s + (0.367 − 0.318i)15-s − 0.509·16-s + (0.415 − 0.719i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.408 + 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.408 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.408 + 0.912i$
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: \((2,\ 819,\ (\ :1/2),\ 0.408 + 0.912i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.833677 - 0.539975i\)
\(L(\frac12)\) \(\approx\) \(0.833677 - 0.539975i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.43 + 0.970i)T \)
7 \( 1 + (0.258 - 0.965i)T \)
13 \( 1 + (-3.08 - 1.87i)T \)
good2 \( 1 + (0.416 + 0.416i)T + 2iT^{2} \)
5 \( 1 + (0.281 - 1.04i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-2.19 + 2.19i)T - 11iT^{2} \)
17 \( 1 + (-1.71 + 2.96i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.73 - 6.46i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-1.55 + 2.69i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 3.34iT - 29T^{2} \)
31 \( 1 + (3.08 + 0.827i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (0.491 - 1.83i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-0.432 + 0.115i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-5.79 + 3.34i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.19 + 4.47i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + 6.74iT - 53T^{2} \)
59 \( 1 + (1.57 - 1.57i)T - 59iT^{2} \)
61 \( 1 + (-0.0157 - 0.0273i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.20 + 11.9i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (10.3 - 2.76i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-9.95 - 9.95i)T + 73iT^{2} \)
79 \( 1 + (-6.31 + 10.9i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-12.2 + 3.27i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (-3.05 - 0.818i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-9.58 - 2.56i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.29370398979316500909622118050, −9.345263500995607619059218096921, −8.500279148239992668378674682766, −7.36015051460976088272016270790, −6.34841014044603294287813935495, −5.93469270407590785073306956134, −4.95958220720450436096692352573, −3.48562665991745318422611023848, −1.99985557701882042410293869878, −0.869685702238445192264252942236, 0.975825413345875012389483275741, 3.22620381327372216702655038805, 4.09614652088136885206718574780, 4.95186581377479669180388663519, 6.14065795305038159510290052238, 6.95181555709927131596352133550, 7.74913050546292564752668756782, 8.977288304690913152548946642371, 9.251547484587836383952007571658, 10.46092044590577945138895835439

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