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Properties

Label 2-819-91.4-c1-0-14
Degree $2$
Conductor $819$
Sign $-0.701 - 0.713i$
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  + 2.58i·2-s − 4.70·4-s + (1.39 − 0.806i)5-s + (1.06 − 2.42i)7-s − 6.99i·8-s + (2.08 + 3.61i)10-s + (−2.34 + 1.35i)11-s + (2.36 + 2.71i)13-s + (6.27 + 2.74i)14-s + 8.69·16-s + 3.12·17-s + (3.18 + 1.84i)19-s + (−6.56 + 3.78i)20-s + (−3.50 − 6.06i)22-s + 1.98·23-s + ⋯
L(s)  = 1  + 1.83i·2-s − 2.35·4-s + (0.624 − 0.360i)5-s + (0.401 − 0.915i)7-s − 2.47i·8-s + (0.659 + 1.14i)10-s + (−0.706 + 0.407i)11-s + (0.656 + 0.753i)13-s + (1.67 + 0.734i)14-s + 2.17·16-s + 0.758·17-s + (0.731 + 0.422i)19-s + (−1.46 + 0.847i)20-s + (−0.746 − 1.29i)22-s + 0.414·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 - 0.713i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.701 - 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.701 - 0.713i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (550, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.701 - 0.713i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.598006 + 1.42647i\)
\(L(\frac12)\) \(\approx\) \(0.598006 + 1.42647i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.06 + 2.42i)T \)
13 \( 1 + (-2.36 - 2.71i)T \)
good2 \( 1 - 2.58iT - 2T^{2} \)
5 \( 1 + (-1.39 + 0.806i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.34 - 1.35i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 - 3.12T + 17T^{2} \)
19 \( 1 + (-3.18 - 1.84i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 1.98T + 23T^{2} \)
29 \( 1 + (2.68 - 4.65i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-9.07 - 5.23i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 5.95iT - 37T^{2} \)
41 \( 1 + (-6.66 - 3.85i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.67 + 2.90i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.913 - 0.527i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.63 + 6.29i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 11.4iT - 59T^{2} \)
61 \( 1 + (-1.46 + 2.53i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-11.7 + 6.79i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.17 - 0.675i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (7.88 + 4.55i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.10 - 5.37i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 2.69iT - 83T^{2} \)
89 \( 1 + 1.75iT - 89T^{2} \)
97 \( 1 + (13.4 - 7.74i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.07515066356870617148414874516, −9.555536070367780606585917098059, −8.517042457336532786905126138554, −7.87616095123894817622136190745, −7.08146300268694464648238958202, −6.33717438281461787284265080518, −5.27674182186396727272264929421, −4.79782566691149469774917873513, −3.59785496679242108535335863886, −1.27838400328866267579680090126, 0.959918025541785863799192756997, 2.43102875079830899643973831151, 2.87645381401872364939381441834, 4.14523072921439433413418308427, 5.39349666068714969360164781627, 5.89911891349862536246512588960, 7.77527255618369644007312125321, 8.549052094210712608232744798955, 9.433469039118928354565410414236, 10.09069190569208130841093879609

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