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Properties

Label 2-819-91.16-c1-0-31
Degree $2$
Conductor $819$
Sign $0.996 - 0.0804i$
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  + 2.37·2-s + 3.63·4-s + (−0.794 + 1.37i)5-s + (2.03 − 1.68i)7-s + 3.87·8-s + (−1.88 + 3.26i)10-s + (1.64 − 2.84i)11-s + (3.29 + 1.46i)13-s + (4.83 − 4.00i)14-s + 1.92·16-s − 5.95·17-s + (3.57 + 6.19i)19-s + (−2.88 + 4.99i)20-s + (3.89 − 6.74i)22-s + 5.54·23-s + ⋯
L(s)  = 1  + 1.67·2-s + 1.81·4-s + (−0.355 + 0.615i)5-s + (0.770 − 0.637i)7-s + 1.36·8-s + (−0.596 + 1.03i)10-s + (0.494 − 0.857i)11-s + (0.913 + 0.406i)13-s + (1.29 − 1.06i)14-s + 0.480·16-s − 1.44·17-s + (0.820 + 1.42i)19-s + (−0.645 + 1.11i)20-s + (0.830 − 1.43i)22-s + 1.15·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0804i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0804i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.996 - 0.0804i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.996 - 0.0804i)\)

Particular Values

\(L(1)\) \(\approx\) \(4.12036 + 0.165923i\)
\(L(\frac12)\) \(\approx\) \(4.12036 + 0.165923i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.03 + 1.68i)T \)
13 \( 1 + (-3.29 - 1.46i)T \)
good2 \( 1 - 2.37T + 2T^{2} \)
5 \( 1 + (0.794 - 1.37i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.64 + 2.84i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + 5.95T + 17T^{2} \)
19 \( 1 + (-3.57 - 6.19i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 5.54T + 23T^{2} \)
29 \( 1 + (3.68 + 6.37i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.24 + 7.35i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 2.55T + 37T^{2} \)
41 \( 1 + (-1.54 - 2.67i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.36 - 9.29i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.29 - 7.43i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.97 + 10.3i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 1.21T + 59T^{2} \)
61 \( 1 + (0.319 + 0.552i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.682 - 1.18i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.19 - 2.06i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (0.309 + 0.535i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.41 + 12.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 10.1T + 83T^{2} \)
89 \( 1 + 8.27T + 89T^{2} \)
97 \( 1 + (-0.831 + 1.44i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.94024422685154371044231412396, −9.479667674466911029782802640644, −8.289382402271389081613196288984, −7.36126474182576297018777148593, −6.49616317837187590779510349374, −5.81702397885572328575426380917, −4.66574588875861005665232860766, −3.86635601116571611968670884476, −3.21349636813256234180988771268, −1.68981430077833210592431191317, 1.67434053565807964066770734774, 2.94009038359837833266995492888, 4.05363973424649301338830863816, 5.00279241478983861927193000252, 5.25202951694077160503182184409, 6.67470713816623561078194499464, 7.19836969217119936330602734643, 8.757188360983016710727757158757, 8.988875383151744934155282834164, 10.83227175121531334785618320165

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