↑

Properties

Label 2-418-11.5-c1-0-12
Degree $2$
Conductor $418$
Sign $0.904 + 0.427i$
Analytic cond. $3.33774$
Root an. cond. $1.82695$
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  + (0.309 + 0.951i)2-s + (−0.128 − 0.0932i)3-s + (−0.809 + 0.587i)4-s + (0.950 − 2.92i)5-s + (0.0490 − 0.150i)6-s + (−1.09 + 0.794i)7-s + (−0.809 − 0.587i)8-s + (−0.919 − 2.82i)9-s + 3.07·10-s + (3.30 + 0.269i)11-s + 0.158·12-s + (−0.162 − 0.500i)13-s + (−1.09 − 0.794i)14-s + (−0.395 + 0.287i)15-s + (0.309 − 0.951i)16-s + (2.23 − 6.88i)17-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (−0.0741 − 0.0538i)3-s + (−0.404 + 0.293i)4-s + (0.425 − 1.30i)5-s + (0.0200 − 0.0616i)6-s + (−0.413 + 0.300i)7-s + (−0.286 − 0.207i)8-s + (−0.306 − 0.943i)9-s + 0.973·10-s + (0.996 + 0.0813i)11-s + 0.0458·12-s + (−0.0451 − 0.138i)13-s + (−0.292 − 0.212i)14-s + (−0.102 + 0.0741i)15-s + (0.0772 − 0.237i)16-s + (0.542 − 1.66i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 + 0.427i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.904 + 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(418\)    =    \(2 \cdot 11 \cdot 19\)
Sign: $0.904 + 0.427i$
Analytic conductor: \(3.33774\)
Root analytic conductor: \(1.82695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{418} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 418,\ (\ :1/2),\ 0.904 + 0.427i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41819 - 0.318130i\)
\(L(\frac12)\) \(\approx\) \(1.41819 - 0.318130i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 - 0.951i)T \)
11 \( 1 + (-3.30 - 0.269i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
good3 \( 1 + (0.128 + 0.0932i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (-0.950 + 2.92i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (1.09 - 0.794i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (0.162 + 0.500i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-2.23 + 6.88i)T + (-13.7 - 9.99i)T^{2} \)
23 \( 1 - 2.09T + 23T^{2} \)
29 \( 1 + (-0.224 + 0.163i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.532 + 1.64i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (7.21 - 5.23i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-6.77 - 4.92i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 6.30T + 43T^{2} \)
47 \( 1 + (1.59 + 1.15i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (0.682 + 2.09i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (7.05 - 5.12i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-1.97 + 6.07i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 1.02T + 67T^{2} \)
71 \( 1 + (1.14 - 3.53i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (1.39 - 1.01i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-2.32 - 7.16i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-2.00 + 6.17i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 10.3T + 89T^{2} \)
97 \( 1 + (3.22 + 9.93i)T + (-78.4 + 57.0i)T^{2} \)
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

−11.50017166658118234054312905264, −9.642895385406707887175554805213, −9.320188868627581615015833764749, −8.566635447715210339040667575353, −7.29610210892450497160389313934, −6.28897806788952767894806369322, −5.45911666438364781946868947669, −4.51756189306774690365455791607, −3.16708357559221451482837801912, −0.967133899490854688149311097465, 1.86021092263016970786930346375, 3.11294027870530687131325412133, 4.06341118681658121024869237249, 5.59395419742531018520048646040, 6.44610686979613111432209883141, 7.44115802029347273138313554373, 8.730642708931577659892653053739, 9.772575374922387871230477470869, 10.72463382849468423757230124480, 10.85935525470753146130250064694

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