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Properties

Label 2-819-117.2-c1-0-47
Degree $2$
Conductor $819$
Sign $-0.942 - 0.334i$
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  + (−1.25 − 1.25i)2-s + (−0.923 − 1.46i)3-s + 1.17i·4-s + (−0.838 + 3.13i)5-s + (−0.682 + 3.00i)6-s + (0.258 − 0.965i)7-s + (−1.04 + 1.04i)8-s + (−1.29 + 2.70i)9-s + (5.00 − 2.88i)10-s + (3.99 − 3.99i)11-s + (1.72 − 1.08i)12-s + (1.66 − 3.19i)13-s + (−1.54 + 0.890i)14-s + (5.36 − 1.66i)15-s + 4.96·16-s + (−0.651 + 1.12i)17-s + ⋯
L(s)  = 1  + (−0.890 − 0.890i)2-s + (−0.533 − 0.846i)3-s + 0.587i·4-s + (−0.375 + 1.40i)5-s + (−0.278 + 1.22i)6-s + (0.0978 − 0.365i)7-s + (−0.367 + 0.367i)8-s + (−0.431 + 0.902i)9-s + (1.58 − 0.913i)10-s + (1.20 − 1.20i)11-s + (0.496 − 0.312i)12-s + (0.461 − 0.887i)13-s + (−0.412 + 0.238i)14-s + (1.38 − 0.429i)15-s + 1.24·16-s + (−0.158 + 0.273i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.942 - 0.334i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.942 - 0.334i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.942 - 0.334i$
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: \((2,\ 819,\ (\ :1/2),\ -0.942 - 0.334i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0693561 + 0.403114i\)
\(L(\frac12)\) \(\approx\) \(0.0693561 + 0.403114i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.923 + 1.46i)T \)
7 \( 1 + (-0.258 + 0.965i)T \)
13 \( 1 + (-1.66 + 3.19i)T \)
good2 \( 1 + (1.25 + 1.25i)T + 2iT^{2} \)
5 \( 1 + (0.838 - 3.13i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-3.99 + 3.99i)T - 11iT^{2} \)
17 \( 1 + (0.651 - 1.12i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.742 + 2.77i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (3.37 - 5.85i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 7.68iT - 29T^{2} \)
31 \( 1 + (8.69 + 2.33i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-0.517 + 1.93i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-0.899 + 0.240i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (1.92 - 1.11i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.84 - 10.6i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + 9.91iT - 53T^{2} \)
59 \( 1 + (0.129 - 0.129i)T - 59iT^{2} \)
61 \( 1 + (5.22 + 9.04i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.31 - 8.63i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (7.00 - 1.87i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (9.23 + 9.23i)T + 73iT^{2} \)
79 \( 1 + (-7.35 + 12.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (9.97 - 2.67i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (15.9 + 4.27i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-8.08 - 2.16i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.04701182079487159840042596722, −8.985788223551683268265592179799, −8.022973732427085191478093329609, −7.38604675194857327386398899158, −6.25243908440342769338044377021, −5.79798675411589222507932118239, −3.76410201477625682177740446755, −2.90318419104252996307330601237, −1.63770462603063701335846267089, −0.31856175483160279479860549800, 1.40191258706743314499982094250, 3.84403956303708324286045626229, 4.47943448539651210200759055528, 5.49905096602809241801016395993, 6.48547254381693224811105854863, 7.25673027742549232976842605847, 8.663929751577447764453323234895, 8.811645413999625893712154673015, 9.483623321241014172987136724260, 10.35073008299398431655319869009

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