↑

Properties

Label 2-416-13.4-c1-0-11
Degree $2$
Conductor $416$
Sign $-0.265 + 0.964i$
Analytic cond. $3.32177$
Root an. cond. $1.82257$
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  + (1.45 − 2.51i)3-s − 2i·5-s + (0.675 − 0.389i)7-s + (−2.73 − 4.73i)9-s + (4.36 + 2.51i)11-s + (−1 + 3.46i)13-s + (−5.03 − 2.90i)15-s + (−3.23 − 5.59i)17-s + (0.675 − 0.389i)19-s − 2.26i·21-s + (−3.58 + 6.20i)23-s + 25-s − 7.16·27-s + (−1.5 + 2.59i)29-s + 1.55i·31-s + ⋯
L(s)  = 1  + (0.839 − 1.45i)3-s − 0.894i·5-s + (0.255 − 0.147i)7-s + (−0.910 − 1.57i)9-s + (1.31 + 0.759i)11-s + (−0.277 + 0.960i)13-s + (−1.30 − 0.751i)15-s + (−0.783 − 1.35i)17-s + (0.154 − 0.0894i)19-s − 0.494i·21-s + (−0.747 + 1.29i)23-s + 0.200·25-s − 1.37·27-s + (−0.278 + 0.482i)29-s + 0.280i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.265 + 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.265 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $-0.265 + 0.964i$
Analytic conductor: \(3.32177\)
Root analytic conductor: \(1.82257\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (225, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :1/2),\ -0.265 + 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11474 - 1.46244i\)
\(L(\frac12)\) \(\approx\) \(1.11474 - 1.46244i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 + (1 - 3.46i)T \)
good3 \( 1 + (-1.45 + 2.51i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + 2iT - 5T^{2} \)
7 \( 1 + (-0.675 + 0.389i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-4.36 - 2.51i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (3.23 + 5.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.675 + 0.389i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.58 - 6.20i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.55iT - 31T^{2} \)
37 \( 1 + (3.69 + 2.13i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.76 + 1.59i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.36 - 7.55i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 7.37iT - 47T^{2} \)
53 \( 1 - 9.46T + 53T^{2} \)
59 \( 1 + (-10.7 + 6.20i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.5 + 2.59i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.675 + 0.389i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-13.0 + 7.55i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 6iT - 73T^{2} \)
79 \( 1 - 10.0T + 79T^{2} \)
83 \( 1 - 10.0iT - 83T^{2} \)
89 \( 1 + (7.96 + 4.59i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.89 - 5.13i)T + (48.5 - 84.0i)T^{2} \)
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

−11.41573353292577166211613175843, −9.435518668552200857355012294545, −9.165183020454601541233588590107, −8.160718122916270441875863541282, −7.11394792244856301572318502555, −6.72369427495300349500842348114, −5.09192932667874622684434748665, −3.87742847627620100571773913165, −2.19619225988201727065988138706, −1.25549760220754058442435390340, 2.46052767009696544430867724731, 3.57978343572820033313778666103, 4.25286770059819553037975205816, 5.67672346210766118670216921156, 6.73629759756520469468812598849, 8.262306326487126236312740801790, 8.683724067135507555557968766374, 9.779135736361888305910173115929, 10.53683893021585835607344477068, 11.03705323814994831204979774996

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