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Properties

Label 2-416-104.101-c1-0-0
Degree $2$
Conductor $416$
Sign $-0.859 - 0.510i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.323 + 0.186i)3-s − 2.04·5-s + (0.768 + 0.443i)7-s + (−1.43 + 2.47i)9-s + (−0.654 − 1.13i)11-s + (−3.57 − 0.490i)13-s + (0.661 − 0.381i)15-s + (−1.02 + 1.76i)17-s + (−3.29 + 5.70i)19-s − 0.331·21-s + (−1.76 − 3.06i)23-s − 0.816·25-s − 2.18i·27-s + (−2.84 + 1.64i)29-s + 7.97i·31-s + ⋯
L(s)  = 1  + (−0.186 + 0.107i)3-s − 0.914·5-s + (0.290 + 0.167i)7-s + (−0.476 + 0.825i)9-s + (−0.197 − 0.341i)11-s + (−0.990 − 0.136i)13-s + (0.170 − 0.0985i)15-s + (−0.247 + 0.429i)17-s + (−0.755 + 1.30i)19-s − 0.0722·21-s + (−0.368 − 0.638i)23-s − 0.163·25-s − 0.421i·27-s + (−0.529 + 0.305i)29-s + 1.43i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.859 - 0.510i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.859 - 0.510i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $-0.859 - 0.510i$
Analytic conductor: \(3.32177\)
Root analytic conductor: \(1.82257\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :1/2),\ -0.859 - 0.510i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.107759 + 0.392189i\)
\(L(\frac12)\) \(\approx\) \(0.107759 + 0.392189i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.57 + 0.490i)T \)
good3 \( 1 + (0.323 - 0.186i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + 2.04T + 5T^{2} \)
7 \( 1 + (-0.768 - 0.443i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.654 + 1.13i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.02 - 1.76i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.29 - 5.70i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.76 + 3.06i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.84 - 1.64i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 7.97iT - 31T^{2} \)
37 \( 1 + (-2.25 - 3.90i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-6.06 + 3.50i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.81 + 3.35i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 0.0334iT - 47T^{2} \)
53 \( 1 + 2.79iT - 53T^{2} \)
59 \( 1 + (-3.94 + 6.82i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-11.4 - 6.62i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.31 - 10.9i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (10.0 + 5.78i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 3.99iT - 73T^{2} \)
79 \( 1 - 1.04T + 79T^{2} \)
83 \( 1 - 11.8T + 83T^{2} \)
89 \( 1 + (12.5 - 7.23i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.592 - 0.342i)T + (48.5 + 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.59537140038712339882458890162, −10.71636623408150912651963091469, −10.02442674387675930329928869876, −8.458809423556299394104348928595, −8.145705262691588985831853125588, −7.04817882451067019317527895100, −5.74606952235436261245102632512, −4.80519661238100177750558412846, −3.69311297514648904253740285431, −2.21058465006549091592255359985, 0.25209605739190198611799124885, 2.48863828395004027914805612209, 3.93019332247272793796256636687, 4.84372935517033923772994542881, 6.15044161945329785448288349171, 7.24675551433309809675193192692, 7.88738185788115738487558053368, 9.085457378090370431233272645683, 9.817903943992353817862354655628, 11.31408820654202236105727610653

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