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Properties

Label 2-819-91.30-c1-0-10
Degree $2$
Conductor $819$
Sign $-0.901 - 0.432i$
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  + (−1.16 + 0.672i)2-s + (−0.0951 + 0.164i)4-s + (3.08 + 1.78i)5-s + (−2.09 + 1.61i)7-s − 2.94i·8-s − 4.79·10-s + 1.27i·11-s + (3.57 + 0.474i)13-s + (1.35 − 3.29i)14-s + (1.79 + 3.10i)16-s + (−3.86 + 6.70i)17-s − 0.943i·19-s + (−0.588 + 0.339i)20-s + (−0.860 − 1.48i)22-s + (−0.823 − 1.42i)23-s + ⋯
L(s)  = 1  + (−0.823 + 0.475i)2-s + (−0.0475 + 0.0824i)4-s + (1.38 + 0.797i)5-s + (−0.792 + 0.610i)7-s − 1.04i·8-s − 1.51·10-s + 0.385i·11-s + (0.991 + 0.131i)13-s + (0.362 − 0.879i)14-s + (0.447 + 0.775i)16-s + (−0.938 + 1.62i)17-s − 0.216i·19-s + (−0.131 + 0.0759i)20-s + (−0.183 − 0.317i)22-s + (−0.171 − 0.297i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.901 - 0.432i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.901 - 0.432i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.901 - 0.432i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (667, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.901 - 0.432i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.204894 + 0.901764i\)
\(L(\frac12)\) \(\approx\) \(0.204894 + 0.901764i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.09 - 1.61i)T \)
13 \( 1 + (-3.57 - 0.474i)T \)
good2 \( 1 + (1.16 - 0.672i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (-3.08 - 1.78i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 - 1.27iT - 11T^{2} \)
17 \( 1 + (3.86 - 6.70i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + 0.943iT - 19T^{2} \)
23 \( 1 + (0.823 + 1.42i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.02 + 3.50i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.46 - 2.57i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.914 + 0.528i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.63 - 2.09i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.91 - 3.31i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.774 - 0.447i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.0399 + 0.0692i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (9.68 + 5.59i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + 7.62T + 61T^{2} \)
67 \( 1 - 6.32iT - 67T^{2} \)
71 \( 1 + (9.89 - 5.71i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (0.658 - 0.380i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.42 + 2.47i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 2.32iT - 83T^{2} \)
89 \( 1 + (6.56 - 3.78i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.414 - 0.239i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.35480684610186397264303353712, −9.580542103601198280550164130313, −9.004325207379997649393550394105, −8.249364899882803891390019359542, −7.01193020450251062169558700518, −6.28074476175231077605539394779, −5.93437782297128330172858572616, −4.17008471554116908280705842317, −2.98832824058091378173359812776, −1.78110929360443965436896081262, 0.60764752380886729882462579210, 1.71587763197124865441463745482, 2.96029146603684671574408234481, 4.53388966940285622066991956049, 5.57178014249277193901533876498, 6.21748645520607060852693359872, 7.41420418271426651052126608155, 8.712201495160582837496475695650, 9.182203207545302168588330650593, 9.696459409956349150185323977059

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