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Properties

Label 2-819-819.275-c1-0-26
Degree $2$
Conductor $819$
Sign $-0.284 - 0.958i$
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  + (−0.534 + 1.99i)2-s + (−0.858 − 1.50i)3-s + (−1.95 − 1.13i)4-s + (0.00724 − 0.0270i)5-s + (3.45 − 0.907i)6-s + (2.08 + 1.62i)7-s + (0.380 − 0.380i)8-s + (−1.52 + 2.58i)9-s + (0.0500 + 0.0288i)10-s + (3.08 − 3.08i)11-s + (−0.0199 + 3.91i)12-s + (−3.51 + 0.800i)13-s + (−4.35 + 3.29i)14-s + (−0.0468 + 0.0123i)15-s + (−1.70 − 2.95i)16-s + 4.58·17-s + ⋯
L(s)  = 1  + (−0.377 + 1.40i)2-s + (−0.495 − 0.868i)3-s + (−0.978 − 0.565i)4-s + (0.00324 − 0.0120i)5-s + (1.41 − 0.370i)6-s + (0.789 + 0.613i)7-s + (0.134 − 0.134i)8-s + (−0.508 + 0.860i)9-s + (0.0158 + 0.00913i)10-s + (0.929 − 0.929i)11-s + (−0.00575 + 1.13i)12-s + (−0.975 + 0.221i)13-s + (−1.16 + 0.881i)14-s + (−0.0121 + 0.00317i)15-s + (−0.426 − 0.738i)16-s + 1.11·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.284 - 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.284 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.284 - 0.958i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (275, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.284 - 0.958i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.637532 + 0.854541i\)
\(L(\frac12)\) \(\approx\) \(0.637532 + 0.854541i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.858 + 1.50i)T \)
7 \( 1 + (-2.08 - 1.62i)T \)
13 \( 1 + (3.51 - 0.800i)T \)
good2 \( 1 + (0.534 - 1.99i)T + (-1.73 - i)T^{2} \)
5 \( 1 + (-0.00724 + 0.0270i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-3.08 + 3.08i)T - 11iT^{2} \)
17 \( 1 - 4.58T + 17T^{2} \)
19 \( 1 + (0.978 + 0.262i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-0.348 + 0.603i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.86 - 2.80i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (5.51 - 5.51i)T - 31iT^{2} \)
37 \( 1 + (1.19 + 1.19i)T + 37iT^{2} \)
41 \( 1 + (-7.38 - 7.38i)T + 41iT^{2} \)
43 \( 1 + 3.29iT - 43T^{2} \)
47 \( 1 + (6.82 - 6.82i)T - 47iT^{2} \)
53 \( 1 + (-9.51 + 5.49i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.80 - 14.1i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (6.31 - 10.9i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.85 + 2.10i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-8.08 + 2.16i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (2.19 - 8.19i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 - 4.15T + 79T^{2} \)
83 \( 1 + (-3.89 + 14.5i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (1.72 - 1.72i)T - 89iT^{2} \)
97 \( 1 + (-2.06 + 2.06i)T - 97iT^{2} \)
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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

−10.46685874451349621498714516502, −9.055597977470546744367762717276, −8.642874250905754922336898240415, −7.70819796994181733186256404674, −7.11242486369179267797766044630, −6.23707487981070701224517359719, −5.52017674716117633345092360010, −4.80038195403840411283777459780, −2.82272075664502222997851366928, −1.20920421697518554581882459971, 0.76760219011499186028029135357, 2.14894818908586282722589596398, 3.51235151481716592510214664625, 4.31343079657261891613645044252, 5.10158147333965924712216262147, 6.46930786343340911278477050916, 7.56972677836142528246616138289, 8.720353384815390027355261933858, 9.668549852543898988592369292170, 10.01067890170628657952503010522

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