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Properties

Label 2-819-91.89-c1-0-16
Degree $2$
Conductor $819$
Sign $0.915 + 0.403i$
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  + (−0.374 − 0.374i)2-s − 1.71i·4-s + (0.545 + 2.03i)5-s + (−2.03 − 1.69i)7-s + (−1.39 + 1.39i)8-s + (0.558 − 0.968i)10-s + (0.745 + 2.78i)11-s + (2.80 + 2.27i)13-s + (0.129 + 1.39i)14-s − 2.39·16-s + 3.29·17-s + (6.53 + 1.75i)19-s + (3.50 − 0.938i)20-s + (0.763 − 1.32i)22-s − 7.84i·23-s + ⋯
L(s)  = 1  + (−0.264 − 0.264i)2-s − 0.859i·4-s + (0.244 + 0.911i)5-s + (−0.769 − 0.638i)7-s + (−0.492 + 0.492i)8-s + (0.176 − 0.306i)10-s + (0.224 + 0.839i)11-s + (0.776 + 0.629i)13-s + (0.0345 + 0.373i)14-s − 0.598·16-s + 0.799·17-s + (1.49 + 0.401i)19-s + (0.783 − 0.209i)20-s + (0.162 − 0.281i)22-s − 1.63i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.915 + 0.403i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.915 + 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.915 + 0.403i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.915 + 0.403i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.32701 - 0.279398i\)
\(L(\frac12)\) \(\approx\) \(1.32701 - 0.279398i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.03 + 1.69i)T \)
13 \( 1 + (-2.80 - 2.27i)T \)
good2 \( 1 + (0.374 + 0.374i)T + 2iT^{2} \)
5 \( 1 + (-0.545 - 2.03i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (-0.745 - 2.78i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 - 3.29T + 17T^{2} \)
19 \( 1 + (-6.53 - 1.75i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + 7.84iT - 23T^{2} \)
29 \( 1 + (0.677 + 1.17i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-6.38 - 1.71i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-2.87 + 2.87i)T - 37iT^{2} \)
41 \( 1 + (6.61 + 1.77i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (4.36 + 2.51i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.53 + 0.947i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-3.87 - 6.71i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.01 - 2.01i)T + 59iT^{2} \)
61 \( 1 + (3.14 - 1.81i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.82 + 0.489i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-9.87 + 2.64i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-1.82 + 6.80i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (1.13 - 1.95i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.97 + 3.97i)T - 83iT^{2} \)
89 \( 1 + (8.73 + 8.73i)T + 89iT^{2} \)
97 \( 1 + (-3.94 - 14.7i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.19133111389611127165977428118, −9.674332573012990907665015243660, −8.728481438461103470103798164461, −7.40286365423421925107850692278, −6.61966499451023507920172450661, −6.05927479731943631054227093433, −4.80092035519214007974271241190, −3.57652548505629294037242733551, −2.46372320635396253342856166574, −1.06110719960534721340967545834, 1.01865943860123625792965642849, 3.05164635507961386316543177964, 3.56462983237539890122194539379, 5.18130841654681314932174285308, 5.85872674222922475632193785688, 6.88783308529466072765582124985, 7.992469228562469532187706392668, 8.525765845708139314481549805218, 9.373687949840776887210001038801, 9.860788339792947776924526022977

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