↑

Properties

Label 2-798-133.10-c1-0-14
Degree $2$
Conductor $798$
Sign $0.0758 + 0.997i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.642 − 0.766i)2-s + (−0.173 − 0.984i)3-s + (−0.173 + 0.984i)4-s + (3.04 − 0.537i)5-s + (−0.642 + 0.766i)6-s + (−2.64 − 0.169i)7-s + (0.866 − 0.500i)8-s + (−0.939 + 0.342i)9-s + (−2.36 − 1.98i)10-s + 4.55·11-s + 12-s + (2.20 + 1.84i)13-s + (1.56 + 2.13i)14-s + (−1.05 − 2.90i)15-s + (−0.939 − 0.342i)16-s + (0.0766 − 0.210i)17-s + ⋯
L(s)  = 1  + (−0.454 − 0.541i)2-s + (−0.100 − 0.568i)3-s + (−0.0868 + 0.492i)4-s + (1.36 − 0.240i)5-s + (−0.262 + 0.312i)6-s + (−0.997 − 0.0642i)7-s + (0.306 − 0.176i)8-s + (−0.313 + 0.114i)9-s + (−0.749 − 0.628i)10-s + 1.37·11-s + 0.288·12-s + (0.610 + 0.512i)13-s + (0.418 + 0.569i)14-s + (−0.273 − 0.750i)15-s + (−0.234 − 0.0855i)16-s + (0.0185 − 0.0510i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0758 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0758 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(798\)    =    \(2 \cdot 3 \cdot 7 \cdot 19\)
Sign: $0.0758 + 0.997i$
Analytic conductor: \(6.37206\)
Root analytic conductor: \(2.52429\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{798} (409, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 798,\ (\ :1/2),\ 0.0758 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06318 - 0.985408i\)
\(L(\frac12)\) \(\approx\) \(1.06318 - 0.985408i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.642 + 0.766i)T \)
3 \( 1 + (0.173 + 0.984i)T \)
7 \( 1 + (2.64 + 0.169i)T \)
19 \( 1 + (-1.53 + 4.08i)T \)
good5 \( 1 + (-3.04 + 0.537i)T + (4.69 - 1.71i)T^{2} \)
11 \( 1 - 4.55T + 11T^{2} \)
13 \( 1 + (-2.20 - 1.84i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-0.0766 + 0.210i)T + (-13.0 - 10.9i)T^{2} \)
23 \( 1 + (1.84 + 1.55i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (-4.41 - 0.778i)T + (27.2 + 9.91i)T^{2} \)
31 \( 1 + (-0.0496 - 0.0859i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-6.04 + 3.49i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.92 + 3.29i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (2.06 + 0.751i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-1.62 - 4.46i)T + (-36.0 + 30.2i)T^{2} \)
53 \( 1 + (8.20 + 1.44i)T + (49.8 + 18.1i)T^{2} \)
59 \( 1 + (11.5 + 4.21i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (3.87 - 4.61i)T + (-10.5 - 60.0i)T^{2} \)
67 \( 1 + (-4.53 + 5.40i)T + (-11.6 - 65.9i)T^{2} \)
71 \( 1 + (0.460 - 1.26i)T + (-54.3 - 45.6i)T^{2} \)
73 \( 1 + (2.13 - 0.376i)T + (68.5 - 24.9i)T^{2} \)
79 \( 1 + (-5.47 + 15.0i)T + (-60.5 - 50.7i)T^{2} \)
83 \( 1 + (13.9 + 8.07i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.93 + 10.9i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (-1.27 - 7.21i)T + (-91.1 + 33.1i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.889695756282428908180815237636, −9.209247391957258034094694671131, −8.855956387431403369317431224580, −7.41142072095759165517281491018, −6.37491573110444113153975577465, −6.10716851208893954505645465072, −4.54780505241170134440283852596, −3.24296714831159984172181526345, −2.08918249915286350105640631425, −1.00837753606408129708530984847, 1.37597805022648275913651424081, 2.96829732022683900697956034562, 4.12061106830692035938977614704, 5.56204091220206369704203077181, 6.17327565535303567088743759786, 6.64572700481639432755988442484, 8.038046953251300165037829699726, 9.064171535742409958020182720127, 9.686582017536904217426530544958, 10.04202762310274777435384744407

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