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Properties

Label 2-819-117.103-c1-0-22
Degree $2$
Conductor $819$
Sign $-0.670 - 0.742i$
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  + (−2.21 + 1.27i)2-s + (1.71 + 0.241i)3-s + (2.26 − 3.91i)4-s + (0.995 + 0.574i)5-s + (−4.10 + 1.65i)6-s + (0.866 − 0.5i)7-s + 6.44i·8-s + (2.88 + 0.827i)9-s − 2.93·10-s + (−3.73 + 2.15i)11-s + (4.82 − 6.17i)12-s + (−0.744 + 3.52i)13-s + (−1.27 + 2.21i)14-s + (1.56 + 1.22i)15-s + (−3.70 − 6.41i)16-s − 2.79·17-s + ⋯
L(s)  = 1  + (−1.56 + 0.902i)2-s + (0.990 + 0.139i)3-s + (1.13 − 1.95i)4-s + (0.445 + 0.257i)5-s + (−1.67 + 0.676i)6-s + (0.327 − 0.188i)7-s + 2.27i·8-s + (0.961 + 0.275i)9-s − 0.928·10-s + (−1.12 + 0.650i)11-s + (1.39 − 1.78i)12-s + (−0.206 + 0.978i)13-s + (−0.341 + 0.591i)14-s + (0.405 + 0.316i)15-s + (−0.926 − 1.60i)16-s − 0.678·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.670 - 0.742i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.670 - 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.670 - 0.742i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.670 - 0.742i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.386921 + 0.870572i\)
\(L(\frac12)\) \(\approx\) \(0.386921 + 0.870572i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.71 - 0.241i)T \)
7 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 + (0.744 - 3.52i)T \)
good2 \( 1 + (2.21 - 1.27i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (-0.995 - 0.574i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (3.73 - 2.15i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + 2.79T + 17T^{2} \)
19 \( 1 + 0.163iT - 19T^{2} \)
23 \( 1 + (3.77 - 6.53i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.90 - 5.03i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.565 - 0.326i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 5.98iT - 37T^{2} \)
41 \( 1 + (-4.39 - 2.53i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.0944 - 0.163i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.08 + 1.77i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 10.5T + 53T^{2} \)
59 \( 1 + (-8.40 - 4.85i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.45 - 7.71i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.80 - 1.61i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 11.6iT - 71T^{2} \)
73 \( 1 + 7.32iT - 73T^{2} \)
79 \( 1 + (-0.0472 - 0.0817i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (10.7 - 6.21i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 11.6iT - 89T^{2} \)
97 \( 1 + (4.86 - 2.80i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.09486394432352234207889276460, −9.619210983868500474015751305559, −8.762641744554195935016674094433, −8.086251619743348095327117231234, −7.30990074369263328732972018195, −6.76123219446197907195052203050, −5.52634640615005253376340634430, −4.33975026387759203289958534128, −2.50057033040353706132321583020, −1.65122515519715056163460690333, 0.68119334740564610302815269580, 2.24840011304476234985248033435, 2.63922002014904447906420961684, 3.97804411884453162373819842593, 5.56120915339453045197674297280, 6.98453831656857702539735699604, 8.058557107582018849231049308062, 8.233312936993647923968924698206, 9.078776975587915260619246629651, 9.911648705413257522360009855957

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