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Properties

Label 2-456-456.11-c1-0-56
Degree $2$
Conductor $456$
Sign $0.302 + 0.953i$
Analytic cond. $3.64117$
Root an. cond. $1.90818$
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  + (0.197 + 1.40i)2-s + (−0.518 − 1.65i)3-s + (−1.92 + 0.552i)4-s + (1.23 − 2.13i)5-s + (2.21 − 1.05i)6-s + 1.87i·7-s + (−1.15 − 2.58i)8-s + (−2.46 + 1.71i)9-s + (3.23 + 1.30i)10-s − 5.59i·11-s + (1.90 + 2.89i)12-s + (−1.69 + 0.979i)13-s + (−2.63 + 0.371i)14-s + (−4.17 − 0.932i)15-s + (3.38 − 2.12i)16-s + (−1.35 − 0.779i)17-s + ⋯
L(s)  = 1  + (0.139 + 0.990i)2-s + (−0.299 − 0.954i)3-s + (−0.961 + 0.276i)4-s + (0.552 − 0.956i)5-s + (0.903 − 0.429i)6-s + 0.710i·7-s + (−0.407 − 0.913i)8-s + (−0.821 + 0.570i)9-s + (1.02 + 0.413i)10-s − 1.68i·11-s + (0.551 + 0.834i)12-s + (−0.470 + 0.271i)13-s + (−0.703 + 0.0991i)14-s + (−1.07 − 0.240i)15-s + (0.847 − 0.531i)16-s + (−0.327 − 0.189i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.302 + 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.302 + 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(456\)    =    \(2^{3} \cdot 3 \cdot 19\)
Sign: $0.302 + 0.953i$
Analytic conductor: \(3.64117\)
Root analytic conductor: \(1.90818\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{456} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 456,\ (\ :1/2),\ 0.302 + 0.953i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.805070 - 0.589306i\)
\(L(\frac12)\) \(\approx\) \(0.805070 - 0.589306i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.197 - 1.40i)T \)
3 \( 1 + (0.518 + 1.65i)T \)
19 \( 1 + (1.76 + 3.98i)T \)
good5 \( 1 + (-1.23 + 2.13i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 - 1.87iT - 7T^{2} \)
11 \( 1 + 5.59iT - 11T^{2} \)
13 \( 1 + (1.69 - 0.979i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.35 + 0.779i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (1.14 + 1.97i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.34 + 5.79i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 8.03iT - 31T^{2} \)
37 \( 1 - 1.00iT - 37T^{2} \)
41 \( 1 + (-8.27 - 4.77i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.11 + 1.92i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.65 + 6.33i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.04 - 10.4i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.68 + 3.28i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.57 - 0.908i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.08 - 10.5i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (7.86 - 13.6i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.76 - 3.05i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.64 - 2.68i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 0.722iT - 83T^{2} \)
89 \( 1 + (-5.15 + 2.97i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.995 + 1.72i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.15969566425056919383891715599, −9.525531830383958592081034679965, −8.767106512669883103540288784037, −8.231468703438265274449356151016, −7.10238452598378540940031124143, −5.86910094287462236803983363491, −5.74416045064188698012233333580, −4.45926875459970024907047701029, −2.55823219121249306561242582130, −0.61120770110634641480819530582, 2.01291224564906886515446168869, 3.31596752253126447842829913922, 4.30645881608662286653225217960, 5.20770382309722957944199135858, 6.42902488079854920601417784284, 7.60171125899388594548498046734, 9.106855563967457933630644195747, 9.830974911116586766604706799123, 10.56172428265789899719791683517, 10.74431091419241889333531906468

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