Properties

Label 2-798-133.59-c1-0-15
Degree $2$
Conductor $798$
Sign $0.492 + 0.870i$
Analytic cond. $6.37206$
Root an. cond. $2.52429$
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  + (−0.342 − 0.939i)2-s + (0.939 − 0.342i)3-s + (−0.766 + 0.642i)4-s + (0.844 − 1.00i)5-s + (−0.642 − 0.766i)6-s + (0.904 + 2.48i)7-s + (0.866 + 0.500i)8-s + (0.766 − 0.642i)9-s + (−1.23 − 0.449i)10-s + (−0.404 − 0.701i)11-s + (−0.500 + 0.866i)12-s + (3.78 − 3.17i)13-s + (2.02 − 1.70i)14-s + (0.449 − 1.23i)15-s + (0.173 − 0.984i)16-s + (−3.86 + 4.61i)17-s + ⋯
L(s)  = 1  + (−0.241 − 0.664i)2-s + (0.542 − 0.197i)3-s + (−0.383 + 0.321i)4-s + (0.377 − 0.450i)5-s + (−0.262 − 0.312i)6-s + (0.341 + 0.939i)7-s + (0.306 + 0.176i)8-s + (0.255 − 0.214i)9-s + (−0.390 − 0.142i)10-s + (−0.122 − 0.211i)11-s + (−0.144 + 0.249i)12-s + (1.04 − 0.879i)13-s + (0.541 − 0.454i)14-s + (0.116 − 0.318i)15-s + (0.0434 − 0.246i)16-s + (−0.938 + 1.11i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.492 + 0.870i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.492 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(798\)    =    \(2 \cdot 3 \cdot 7 \cdot 19\)
Sign: $0.492 + 0.870i$
Analytic conductor: \(6.37206\)
Root analytic conductor: \(2.52429\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{798} (325, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 798,\ (\ :1/2),\ 0.492 + 0.870i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.58000 - 0.921409i\)
\(L(\frac12)\) \(\approx\) \(1.58000 - 0.921409i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.342 + 0.939i)T \)
3 \( 1 + (-0.939 + 0.342i)T \)
7 \( 1 + (-0.904 - 2.48i)T \)
19 \( 1 + (-3.99 + 1.74i)T \)
good5 \( 1 + (-0.844 + 1.00i)T + (-0.868 - 4.92i)T^{2} \)
11 \( 1 + (0.404 + 0.701i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.78 + 3.17i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (3.86 - 4.61i)T + (-2.95 - 16.7i)T^{2} \)
23 \( 1 + (0.564 + 3.20i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (-5.40 + 0.952i)T + (27.2 - 9.91i)T^{2} \)
31 \( 1 - 8.86T + 31T^{2} \)
37 \( 1 + (-6.12 + 3.53i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (7.12 + 5.98i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (5.98 - 2.17i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (-1.26 - 1.51i)T + (-8.16 + 46.2i)T^{2} \)
53 \( 1 + (7.20 + 8.59i)T + (-9.20 + 52.1i)T^{2} \)
59 \( 1 + (-6.20 - 5.20i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (2.48 - 0.437i)T + (57.3 - 20.8i)T^{2} \)
67 \( 1 + (-2.41 + 6.62i)T + (-51.3 - 43.0i)T^{2} \)
71 \( 1 + (-3.21 - 8.83i)T + (-54.3 + 45.6i)T^{2} \)
73 \( 1 + (0.0625 + 0.171i)T + (-55.9 + 46.9i)T^{2} \)
79 \( 1 + (12.3 + 2.17i)T + (74.2 + 27.0i)T^{2} \)
83 \( 1 + (-10.9 + 6.30i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (8.34 + 3.03i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (3.23 - 18.3i)T + (-91.1 - 33.1i)T^{2} \)
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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

−10.08582849596135639335142166251, −9.185715659482450246302198433741, −8.383160763218812861440098862666, −8.188255006061520855597139304640, −6.59108719339119922794874888290, −5.63728690136304630594433208289, −4.61259044987252098529294126412, −3.33380476276232495960747906188, −2.36928687250124121314441964258, −1.19108784620780584208883877439, 1.35370122337129681848892160610, 2.91798545734989548144482041844, 4.19531119967162459672073287972, 4.94411594014454501431853610546, 6.40847067000037361756681967194, 6.88094062241047740614904249298, 7.903658572927887007478432672362, 8.574138458725337819581426767797, 9.651145466422991219899773575904, 10.07816080118752272967188902952

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