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Properties

Label 2-819-91.30-c1-0-34
Degree $2$
Conductor $819$
Sign $0.472 + 0.881i$
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  + (−0.356 + 0.205i)2-s + (−0.915 + 1.58i)4-s + (1.54 + 0.892i)5-s + (0.355 − 2.62i)7-s − 1.57i·8-s − 0.734·10-s − 5.02i·11-s + (−0.0961 − 3.60i)13-s + (0.412 + 1.00i)14-s + (−1.50 − 2.60i)16-s + (−3.96 + 6.86i)17-s − 3.16i·19-s + (−2.82 + 1.63i)20-s + (1.03 + 1.79i)22-s + (−3.94 − 6.83i)23-s + ⋯
L(s)  = 1  + (−0.252 + 0.145i)2-s + (−0.457 + 0.792i)4-s + (0.691 + 0.398i)5-s + (0.134 − 0.990i)7-s − 0.557i·8-s − 0.232·10-s − 1.51i·11-s + (−0.0266 − 0.999i)13-s + (0.110 + 0.269i)14-s + (−0.376 − 0.652i)16-s + (−0.961 + 1.66i)17-s − 0.725i·19-s + (−0.632 + 0.365i)20-s + (0.220 + 0.382i)22-s + (−0.822 − 1.42i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.472 + 0.881i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.472 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.472 + 0.881i$
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: \((2,\ 819,\ (\ :1/2),\ 0.472 + 0.881i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.904785 - 0.541857i\)
\(L(\frac12)\) \(\approx\) \(0.904785 - 0.541857i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.355 + 2.62i)T \)
13 \( 1 + (0.0961 + 3.60i)T \)
good2 \( 1 + (0.356 - 0.205i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (-1.54 - 0.892i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + 5.02iT - 11T^{2} \)
17 \( 1 + (3.96 - 6.86i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + 3.16iT - 19T^{2} \)
23 \( 1 + (3.94 + 6.83i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.23 - 3.86i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.24 + 2.44i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-7.15 + 4.13i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.53 - 1.46i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.50 + 2.60i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.196 + 0.113i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.51 - 4.35i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.05 - 0.608i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 - 0.256T + 61T^{2} \)
67 \( 1 - 6.41iT - 67T^{2} \)
71 \( 1 + (3.64 - 2.10i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-4.59 + 2.65i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.43 + 9.41i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 14.0iT - 83T^{2} \)
89 \( 1 + (5.18 - 2.99i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.99 - 2.30i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.34450339540455424203394853106, −9.059883603946338571059862103577, −8.346863377912731009815136736869, −7.74662563713818680375203763259, −6.55151145611202297233729421868, −5.95836482079331856515533459862, −4.47838173601221046114503643911, −3.66229146651180055083451803988, −2.55122438713417606266634379581, −0.57030573695366149863079740768, 1.65575954159695451630825202192, 2.31775586504606342219187343309, 4.35603085565862799186348018656, 5.06163234282398134485936543636, 5.85034798748114196675354430720, 6.82174608021872499176807063528, 7.988745780654312759125906212383, 9.105923998265291607694789242032, 9.639260663380552331444529709966, 9.823673615369325612128417272831

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