↑

Properties

Label 2-416-104.61-c1-0-2
Degree $2$
Conductor $416$
Sign $0.995 - 0.0966i$
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  + (−1.39 − 0.802i)3-s − 0.556i·5-s + (2.30 + 3.98i)7-s + (−0.210 − 0.364i)9-s + (−1.09 − 0.634i)11-s + (2.69 + 2.39i)13-s + (−0.446 + 0.773i)15-s + (1.16 + 2.01i)17-s + (4.57 − 2.63i)19-s − 7.39i·21-s + (0.778 − 1.34i)23-s + 4.69·25-s + 5.49i·27-s + (0.923 + 0.533i)29-s + 7.00·31-s + ⋯
L(s)  = 1  + (−0.802 − 0.463i)3-s − 0.248i·5-s + (0.870 + 1.50i)7-s + (−0.0702 − 0.121i)9-s + (−0.331 − 0.191i)11-s + (0.748 + 0.662i)13-s + (−0.115 + 0.199i)15-s + (0.281 + 0.488i)17-s + (1.04 − 0.605i)19-s − 1.61i·21-s + (0.162 − 0.281i)23-s + 0.938·25-s + 1.05i·27-s + (0.171 + 0.0989i)29-s + 1.25·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $0.995 - 0.0966i$
Analytic conductor: \(3.32177\)
Root analytic conductor: \(1.82257\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :1/2),\ 0.995 - 0.0966i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16859 + 0.0566144i\)
\(L(\frac12)\) \(\approx\) \(1.16859 + 0.0566144i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.69 - 2.39i)T \)
good3 \( 1 + (1.39 + 0.802i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + 0.556iT - 5T^{2} \)
7 \( 1 + (-2.30 - 3.98i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.09 + 0.634i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.16 - 2.01i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.57 + 2.63i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.778 + 1.34i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.923 - 0.533i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 7.00T + 31T^{2} \)
37 \( 1 + (2.43 + 1.40i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.44 + 5.97i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.72 + 3.30i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 6.29T + 47T^{2} \)
53 \( 1 - 11.3iT - 53T^{2} \)
59 \( 1 + (12.1 - 6.99i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.41 - 1.97i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.16 - 1.25i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.152 + 0.264i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 7.67T + 73T^{2} \)
79 \( 1 + 8.75T + 79T^{2} \)
83 \( 1 + 6.78iT - 83T^{2} \)
89 \( 1 + (-6.28 + 10.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.36 - 2.36i)T + (-48.5 + 84.0i)T^{2} \)
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

−11.43893259738048863048458052574, −10.63361354694169541460023374503, −9.053627606406451848601484379479, −8.708982110815851608599123022016, −7.48311030996298056709243478120, −6.24417077034598165520966232678, −5.60903044668816583546853919697, −4.68321014603283190829189185547, −2.86790312886735058195534177253, −1.31741633309588119883124515436, 1.06649531124839614676191411905, 3.22778860377910884099503592289, 4.53836602543970837121057760442, 5.22369594203957962364573840036, 6.42193392766086160806085212815, 7.60600520049479774989986452994, 8.158699420914772971903275618229, 9.808005726791562314614869125064, 10.45249620315874123834931883225, 11.09193161666603660567768289368

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