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Properties

Label 2-456-24.11-c1-0-31
Degree $2$
Conductor $456$
Sign $0.999 + 0.0330i$
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  + (1.40 − 0.183i)2-s + (−1.57 − 0.711i)3-s + (1.93 − 0.514i)4-s + 2.07·5-s + (−2.34 − 0.708i)6-s + 4.67i·7-s + (2.61 − 1.07i)8-s + (1.98 + 2.24i)9-s + (2.90 − 0.380i)10-s − 0.907i·11-s + (−3.41 − 0.562i)12-s + 0.234i·13-s + (0.858 + 6.56i)14-s + (−3.27 − 1.47i)15-s + (3.47 − 1.98i)16-s − 1.17i·17-s + ⋯
L(s)  = 1  + (0.991 − 0.129i)2-s + (−0.911 − 0.410i)3-s + (0.966 − 0.257i)4-s + 0.926·5-s + (−0.957 − 0.289i)6-s + 1.76i·7-s + (0.924 − 0.380i)8-s + (0.662 + 0.749i)9-s + (0.918 − 0.120i)10-s − 0.273i·11-s + (−0.986 − 0.162i)12-s + 0.0649i·13-s + (0.229 + 1.75i)14-s + (−0.844 − 0.380i)15-s + (0.867 − 0.497i)16-s − 0.284i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0330i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0330i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(456\)    =    \(2^{3} \cdot 3 \cdot 19\)
Sign: $0.999 + 0.0330i$
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: \((2,\ 456,\ (\ :1/2),\ 0.999 + 0.0330i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.30179 - 0.0380702i\)
\(L(\frac12)\) \(\approx\) \(2.30179 - 0.0380702i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 + 0.183i)T \)
3 \( 1 + (1.57 + 0.711i)T \)
19 \( 1 + T \)
good5 \( 1 - 2.07T + 5T^{2} \)
7 \( 1 - 4.67iT - 7T^{2} \)
11 \( 1 + 0.907iT - 11T^{2} \)
13 \( 1 - 0.234iT - 13T^{2} \)
17 \( 1 + 1.17iT - 17T^{2} \)
23 \( 1 + 0.116T + 23T^{2} \)
29 \( 1 - 7.50T + 29T^{2} \)
31 \( 1 + 4.34iT - 31T^{2} \)
37 \( 1 + 10.2iT - 37T^{2} \)
41 \( 1 - 4.24iT - 41T^{2} \)
43 \( 1 + 7.94T + 43T^{2} \)
47 \( 1 + 7.00T + 47T^{2} \)
53 \( 1 - 0.108T + 53T^{2} \)
59 \( 1 - 12.9iT - 59T^{2} \)
61 \( 1 - 2.38iT - 61T^{2} \)
67 \( 1 - 2.38T + 67T^{2} \)
71 \( 1 - 2.06T + 71T^{2} \)
73 \( 1 + 10.2T + 73T^{2} \)
79 \( 1 - 9.23iT - 79T^{2} \)
83 \( 1 + 8.40iT - 83T^{2} \)
89 \( 1 + 16.5iT - 89T^{2} \)
97 \( 1 + 6.29T + 97T^{2} \)
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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

−11.47751058816026812313702625846, −10.39379493676112613854352460491, −9.503637150041145533691510613443, −8.246767391919926284951056657247, −6.87698040994389779755530489628, −5.93070871024334779500140728498, −5.65252211624456695859994630670, −4.62061021827506775897581955230, −2.74077593961375757061215782409, −1.81421695605559869092568793600, 1.45633033675159975999806574196, 3.42418859517802802983121818943, 4.47531363420354161216215198725, 5.18253433482952957579961484894, 6.49072350271092181905543334398, 6.77461335174813752663058362861, 8.065457464286510981506658865732, 9.891220674359328825391262132137, 10.28860952861030914992908183935, 11.03413501982320184053652252000

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