↑

Properties

Label 2-351-117.4-c1-0-0
Degree $2$
Conductor $351$
Sign $-0.815 - 0.578i$
Analytic cond. $2.80274$
Root an. cond. $1.67414$
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  + 1.73i·2-s − 0.999·4-s + (−1.5 − 0.866i)5-s + (1.5 + 0.866i)7-s + 1.73i·8-s + (1.49 − 2.59i)10-s + 3.46i·11-s + (−1 + 3.46i)13-s + (−1.49 + 2.59i)14-s − 5·16-s + (1.5 + 2.59i)17-s + (−1.5 + 0.866i)19-s + (1.49 + 0.866i)20-s − 5.99·22-s + (−1.5 − 2.59i)23-s + ⋯
L(s)  = 1  + 1.22i·2-s − 0.499·4-s + (−0.670 − 0.387i)5-s + (0.566 + 0.327i)7-s + 0.612i·8-s + (0.474 − 0.821i)10-s + 1.04i·11-s + (−0.277 + 0.960i)13-s + (−0.400 + 0.694i)14-s − 1.25·16-s + (0.363 + 0.630i)17-s + (−0.344 + 0.198i)19-s + (0.335 + 0.193i)20-s − 1.27·22-s + (−0.312 − 0.541i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.815 - 0.578i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.815 - 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(351\)    =    \(3^{3} \cdot 13\)
Sign: $-0.815 - 0.578i$
Analytic conductor: \(2.80274\)
Root analytic conductor: \(1.67414\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{351} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 351,\ (\ :1/2),\ -0.815 - 0.578i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.369228 + 1.15808i\)
\(L(\frac12)\) \(\approx\) \(0.369228 + 1.15808i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 + (1 - 3.46i)T \)
good2 \( 1 - 1.73iT - 2T^{2} \)
5 \( 1 + (1.5 + 0.866i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.5 - 0.866i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 - 3.46iT - 11T^{2} \)
17 \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.5 - 0.866i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.5 + 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + (-7.5 - 4.33i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.5 + 2.59i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-10.5 + 6.06i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.5 - 2.59i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 - 3.46iT - 59T^{2} \)
61 \( 1 + (-2.5 + 4.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.5 + 6.06i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-7.5 + 4.33i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 6.92iT - 73T^{2} \)
79 \( 1 + (-5.5 - 9.52i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.5 - 2.59i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (13.5 + 7.79i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-13.5 - 7.79i)T + (48.5 + 84.0i)T^{2} \)
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

−12.06619737016411807785522041212, −11.02005849740713803305811928789, −9.812346229064902010766836851281, −8.563496860371438946499828776837, −8.103600574925977134578177387429, −7.09234667700520301515797563646, −6.26171339371502801068776451100, −4.96217456048791717203231849615, −4.28082986438875193696744590204, −2.12649420018439191719247805159, 0.877707640178488843338532355375, 2.71845209098039719985589477093, 3.56513132577315024591706663239, 4.77330383031456677196663716152, 6.25930314896276712778996751898, 7.53600819442669166621071450605, 8.286376502533096972212903611671, 9.645936885415560964816513940506, 10.44496939017180229589533505570, 11.31054202148972524382400827311

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