↑

Properties

Label 2-798-133.122-c1-0-7
Degree $2$
Conductor $798$
Sign $-0.140 - 0.990i$
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  − i·2-s + (0.5 + 0.866i)3-s − 4-s + 2.33i·5-s + (0.866 − 0.5i)6-s + (−0.397 + 2.61i)7-s + i·8-s + (−0.499 + 0.866i)9-s + 2.33·10-s + (1.22 − 2.12i)11-s + (−0.5 − 0.866i)12-s + (−0.232 − 0.403i)13-s + (2.61 + 0.397i)14-s + (−2.02 + 1.16i)15-s + 16-s + (−2.86 + 1.65i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.288 + 0.499i)3-s − 0.5·4-s + 1.04i·5-s + (0.353 − 0.204i)6-s + (−0.150 + 0.988i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s + 0.739·10-s + (0.369 − 0.640i)11-s + (−0.144 − 0.249i)12-s + (−0.0645 − 0.111i)13-s + (0.699 + 0.106i)14-s + (−0.522 + 0.301i)15-s + 0.250·16-s + (−0.695 + 0.401i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.140 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.140 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(798\)    =    \(2 \cdot 3 \cdot 7 \cdot 19\)
Sign: $-0.140 - 0.990i$
Analytic conductor: \(6.37206\)
Root analytic conductor: \(2.52429\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{798} (787, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 798,\ (\ :1/2),\ -0.140 - 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.766287 + 0.883111i\)
\(L(\frac12)\) \(\approx\) \(0.766287 + 0.883111i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (0.397 - 2.61i)T \)
19 \( 1 + (2.42 + 3.61i)T \)
good5 \( 1 - 2.33iT - 5T^{2} \)
11 \( 1 + (-1.22 + 2.12i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.232 + 0.403i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.86 - 1.65i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (4.10 - 7.10i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.924 + 0.533i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.97 - 8.61i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.95 - 1.13i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.87 - 8.44i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.54 + 6.14i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-9.42 - 5.43i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 6.34iT - 53T^{2} \)
59 \( 1 + (1.34 + 2.33i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-8.39 - 4.84i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + 5.49iT - 67T^{2} \)
71 \( 1 + (5.52 + 3.19i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-13.9 + 8.04i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 9.03iT - 79T^{2} \)
83 \( 1 - 0.460iT - 83T^{2} \)
89 \( 1 + (7.44 - 12.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.04 + 10.4i)T + (-48.5 - 84.0i)T^{2} \)
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

−10.65528386071194671726146143761, −9.692351775935003994121605771931, −8.960237303929847751389157274081, −8.293783567562519600968007614527, −6.98665703253312604187989335702, −6.04375160110013497057454983393, −5.05138683929080685349957394782, −3.74321122221138535236551257028, −3.02009500567035976689899641912, −2.03113262418299230130559751434, 0.55018244478933490837972588996, 2.04820713894193485518780610320, 3.98682565852038483254919834676, 4.48543172770860789122437408278, 5.74729589035122122603247766928, 6.72616625418839503254832449939, 7.40654014583765271031595596316, 8.303534281740773760182977531087, 8.960265111643945209074442180172, 9.817762865201442073769868730778

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