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Properties

Label 2-456-456.251-c1-0-43
Degree $2$
Conductor $456$
Sign $0.169 + 0.985i$
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  + (0.483 − 1.32i)2-s + (−0.724 + 1.57i)3-s + (−1.53 − 1.28i)4-s + (1.74 + 1.72i)6-s + (−2.44 + 1.41i)8-s + (−1.94 − 2.28i)9-s + (5.54 − 3.19i)11-s + (3.13 − 1.47i)12-s + (0.694 + 3.93i)16-s + (2.74 − 7.54i)17-s + (−3.97 + 1.48i)18-s + (3.49 − 2.60i)19-s + (−1.57 − 8.91i)22-s + (−0.449 − 4.87i)24-s + (−0.868 + 4.92i)25-s + ⋯
L(s)  = 1  + (0.342 − 0.939i)2-s + (−0.418 + 0.908i)3-s + (−0.766 − 0.642i)4-s + (0.710 + 0.703i)6-s + (−0.866 + 0.499i)8-s + (−0.649 − 0.760i)9-s + (1.67 − 0.964i)11-s + (0.904 − 0.426i)12-s + (0.173 + 0.984i)16-s + (0.665 − 1.82i)17-s + (−0.936 + 0.350i)18-s + (0.801 − 0.598i)19-s + (−0.335 − 1.90i)22-s + (−0.0917 − 0.995i)24-s + (−0.173 + 0.984i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.169 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.169 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(456\)    =    \(2^{3} \cdot 3 \cdot 19\)
Sign: $0.169 + 0.985i$
Analytic conductor: \(3.64117\)
Root analytic conductor: \(1.90818\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{456} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 456,\ (\ :1/2),\ 0.169 + 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02932 - 0.867256i\)
\(L(\frac12)\) \(\approx\) \(1.02932 - 0.867256i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.483 + 1.32i)T \)
3 \( 1 + (0.724 - 1.57i)T \)
19 \( 1 + (-3.49 + 2.60i)T \)
good5 \( 1 + (0.868 - 4.92i)T^{2} \)
7 \( 1 + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-5.54 + 3.19i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (12.2 + 4.44i)T^{2} \)
17 \( 1 + (-2.74 + 7.54i)T + (-13.0 - 10.9i)T^{2} \)
23 \( 1 + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + (12.3 - 2.17i)T + (38.5 - 14.0i)T^{2} \)
43 \( 1 + (-9.45 + 7.93i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (2.86 - 7.86i)T + (-45.1 - 37.9i)T^{2} \)
61 \( 1 + (-10.5 - 60.0i)T^{2} \)
67 \( 1 + (4.95 - 1.80i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (-2.96 - 16.8i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (74.2 - 27.0i)T^{2} \)
83 \( 1 + (-8.14 - 4.70i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (12.5 + 2.21i)T + (83.6 + 30.4i)T^{2} \)
97 \( 1 + (-17.4 - 6.35i)T + (74.3 + 62.3i)T^{2} \)
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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

−11.15155199670518067978549949025, −9.979164856651615264970697784131, −9.322913299610551641153277554220, −8.748799336567587464111870357343, −6.95141327934809892810213311345, −5.73145017515915542247472441247, −4.99228955681794841692004119760, −3.80534257862857159200608861055, −3.06346133532831976093863233943, −0.920972198179018349195609082061, 1.55174972560980987547613765434, 3.58701735377660120286751990238, 4.70233908074462032414536175340, 6.02911121405473533913248001187, 6.44446865585006123020622881424, 7.49805657133615768048522923437, 8.198152671237099817976996269504, 9.244240462300729414648613342394, 10.29327190494512832166265547656, 11.72807760516454853515730672754

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