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Properties

Label 2-456-24.11-c1-0-21
Degree $2$
Conductor $456$
Sign $0.790 + 0.612i$
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  + (−1.16 − 0.796i)2-s + (−0.732 + 1.56i)3-s + (0.732 + 1.86i)4-s − 4.19·5-s + (2.10 − 1.25i)6-s − 1.07i·7-s + (0.625 − 2.75i)8-s + (−1.92 − 2.29i)9-s + (4.90 + 3.33i)10-s + 0.479i·11-s + (−3.45 − 0.213i)12-s − 0.199i·13-s + (−0.855 + 1.25i)14-s + (3.07 − 6.58i)15-s + (−2.92 + 2.72i)16-s + 1.44i·17-s + ⋯
L(s)  = 1  + (−0.826 − 0.562i)2-s + (−0.422 + 0.906i)3-s + (0.366 + 0.930i)4-s − 1.87·5-s + (0.859 − 0.510i)6-s − 0.406i·7-s + (0.221 − 0.975i)8-s + (−0.642 − 0.766i)9-s + (1.55 + 1.05i)10-s + 0.144i·11-s + (−0.998 − 0.0616i)12-s − 0.0552i·13-s + (−0.228 + 0.335i)14-s + (0.792 − 1.69i)15-s + (−0.731 + 0.681i)16-s + 0.350i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.790 + 0.612i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.790 + 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(456\)    =    \(2^{3} \cdot 3 \cdot 19\)
Sign: $0.790 + 0.612i$
Analytic conductor: \(3.64117\)
Root analytic conductor: \(1.90818\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{456} (419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 456,\ (\ :1/2),\ 0.790 + 0.612i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.424102 - 0.145179i\)
\(L(\frac12)\) \(\approx\) \(0.424102 - 0.145179i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.16 + 0.796i)T \)
3 \( 1 + (0.732 - 1.56i)T \)
19 \( 1 - T \)
good5 \( 1 + 4.19T + 5T^{2} \)
7 \( 1 + 1.07iT - 7T^{2} \)
11 \( 1 - 0.479iT - 11T^{2} \)
13 \( 1 + 0.199iT - 13T^{2} \)
17 \( 1 - 1.44iT - 17T^{2} \)
23 \( 1 - 4.44T + 23T^{2} \)
29 \( 1 - 3.42T + 29T^{2} \)
31 \( 1 + 7.13iT - 31T^{2} \)
37 \( 1 - 7.53iT - 37T^{2} \)
41 \( 1 + 4.55iT - 41T^{2} \)
43 \( 1 + 1.12T + 43T^{2} \)
47 \( 1 + 2.57T + 47T^{2} \)
53 \( 1 - 9.25T + 53T^{2} \)
59 \( 1 + 9.13iT - 59T^{2} \)
61 \( 1 + 14.3iT - 61T^{2} \)
67 \( 1 + 15.4T + 67T^{2} \)
71 \( 1 - 6.76T + 71T^{2} \)
73 \( 1 + 3.91T + 73T^{2} \)
79 \( 1 + 10.0iT - 79T^{2} \)
83 \( 1 + 12.5iT - 83T^{2} \)
89 \( 1 - 14.3iT - 89T^{2} \)
97 \( 1 + 5.39T + 97T^{2} \)
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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

−10.91752326831822773479119718533, −10.32026771265350843121894975822, −9.216608548326396733337121481588, −8.362820446698442491748893891219, −7.58688694054538995967692951616, −6.62021558614184567378696881243, −4.76727669849035972883429946297, −3.92779876111644414001745300076, −3.14081277952275074719640245300, −0.55218671981302193737961955421, 0.920858359826773122232138888148, 2.87624826418164337043529146350, 4.64532919483586254459189811192, 5.73018304827284257201421883512, 7.08660585406095390634960475963, 7.29142830004675646143461393420, 8.411701371572168640161381007191, 8.830399653744239225234792202372, 10.47085677281233268278719876896, 11.25500600641107428430194955211

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