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Properties

Label 2-798-7.4-c1-0-0
Degree $2$
Conductor $798$
Sign $-0.877 - 0.478i$
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  + (0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (−1.05 − 1.83i)5-s + 0.999·6-s + (−1.11 + 2.39i)7-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (1.05 − 1.83i)10-s + (−1.67 + 2.90i)11-s + (0.499 + 0.866i)12-s − 4.23·13-s + (−2.63 + 0.231i)14-s − 2.11·15-s + (−0.5 − 0.866i)16-s + (−3.47 + 6.02i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (−0.473 − 0.820i)5-s + 0.408·6-s + (−0.422 + 0.906i)7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (0.334 − 0.580i)10-s + (−0.505 + 0.875i)11-s + (0.144 + 0.249i)12-s − 1.17·13-s + (−0.704 + 0.0619i)14-s − 0.546·15-s + (−0.125 − 0.216i)16-s + (−0.843 + 1.46i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.877 - 0.478i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.877 - 0.478i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(798\)    =    \(2 \cdot 3 \cdot 7 \cdot 19\)
Sign: $-0.877 - 0.478i$
Analytic conductor: \(6.37206\)
Root analytic conductor: \(2.52429\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{798} (571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 798,\ (\ :1/2),\ -0.877 - 0.478i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.189267 + 0.742430i\)
\(L(\frac12)\) \(\approx\) \(0.189267 + 0.742430i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (1.11 - 2.39i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (1.05 + 1.83i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.67 - 2.90i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 4.23T + 13T^{2} \)
17 \( 1 + (3.47 - 6.02i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-3.59 - 6.21i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 7.29T + 29T^{2} \)
31 \( 1 + (2.27 - 3.93i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.126 - 0.218i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 10.3T + 41T^{2} \)
43 \( 1 - 3.17T + 43T^{2} \)
47 \( 1 + (3.57 + 6.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.878 - 1.52i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.52 - 11.2i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.45 + 5.99i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.15 + 3.72i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 10.4T + 71T^{2} \)
73 \( 1 + (-6.38 + 11.0i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.13 + 1.96i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 16.5T + 83T^{2} \)
89 \( 1 + (1.89 + 3.28i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 6.18T + 97T^{2} \)
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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

−10.54193575443870570066538139281, −9.450000076511385056805963691009, −8.730968900077356071820552843764, −8.070803482241944853841641297711, −7.16713133365417573396281182373, −6.36920321861247214416480843157, −5.20106599863677353741189142792, −4.58545504901171073801109736671, −3.20757446524576624713459809430, −1.97606054308563600801443743768, 0.30720162304670363033147764612, 2.68118609878816025019581372597, 3.14230253370120729577936608194, 4.34695367096310123993979429012, 5.05702266793021028543682566457, 6.54582492670127599472189518226, 7.22025897467614914382949613941, 8.266879638054880949791702768327, 9.352522325166299739428565885186, 10.09706828266556924218147883891

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