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Properties

Label 2-819-91.24-c1-0-19
Degree $2$
Conductor $819$
Sign $0.379 + 0.925i$
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  + (−0.251 + 0.0673i)2-s + (−1.67 + 0.966i)4-s + (−3.35 − 0.900i)5-s + (0.422 + 2.61i)7-s + (0.723 − 0.723i)8-s + 0.905·10-s + (−1.69 + 1.69i)11-s + (−1.60 + 3.23i)13-s + (−0.282 − 0.628i)14-s + (1.79 − 3.11i)16-s + (−1.67 − 2.89i)17-s + (4.43 − 4.43i)19-s + (6.49 − 1.73i)20-s + (0.312 − 0.541i)22-s + (−0.136 − 0.0787i)23-s + ⋯
L(s)  = 1  + (−0.177 + 0.0476i)2-s + (−0.836 + 0.483i)4-s + (−1.50 − 0.402i)5-s + (0.159 + 0.987i)7-s + (0.255 − 0.255i)8-s + 0.286·10-s + (−0.512 + 0.512i)11-s + (−0.443 + 0.896i)13-s + (−0.0754 − 0.167i)14-s + (0.449 − 0.779i)16-s + (−0.405 − 0.702i)17-s + (1.01 − 1.01i)19-s + (1.45 − 0.388i)20-s + (0.0666 − 0.115i)22-s + (−0.0284 − 0.0164i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.379 + 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.379 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.379 + 0.925i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (388, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.379 + 0.925i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.355552 - 0.238434i\)
\(L(\frac12)\) \(\approx\) \(0.355552 - 0.238434i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.422 - 2.61i)T \)
13 \( 1 + (1.60 - 3.23i)T \)
good2 \( 1 + (0.251 - 0.0673i)T + (1.73 - i)T^{2} \)
5 \( 1 + (3.35 + 0.900i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (1.69 - 1.69i)T - 11iT^{2} \)
17 \( 1 + (1.67 + 2.89i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.43 + 4.43i)T - 19iT^{2} \)
23 \( 1 + (0.136 + 0.0787i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.530 + 0.919i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.37 + 5.14i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (0.635 + 2.37i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-11.0 - 2.95i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (6.74 + 3.89i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.03 + 7.60i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-3.19 + 5.54i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.931 - 3.47i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + 12.4iT - 61T^{2} \)
67 \( 1 + (-3.01 - 3.01i)T + 67iT^{2} \)
71 \( 1 + (3.09 - 0.829i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-7.36 + 1.97i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-3.17 - 5.49i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (10.4 - 10.4i)T - 83iT^{2} \)
89 \( 1 + (-13.8 + 3.71i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (1.40 + 5.24i)T + (-84.0 + 48.5i)T^{2} \)
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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

−9.675498518696430515329568880631, −9.168794789330046572324409107971, −8.388737972453606835665216146130, −7.64219362728768193784959282602, −7.00206530585377056633352656415, −5.24930769251756299360896598585, −4.67797252336603948121398232640, −3.78820260257304889385266963889, −2.52737796978283562858291617112, −0.30048892717264824392736025653, 1.00577651937052043234349545926, 3.22151375976990477350315715252, 3.98542369802462213713510500878, 4.87680522537143087056748804785, 5.93752355673746406865904658827, 7.32160483415782850177866492905, 7.86519762089536582946343979218, 8.472229212785270670724584479466, 9.675689484507986841377112707788, 10.68269749315572710658867493298

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