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Properties

Label 2-456-456.11-c1-0-73
Degree $2$
Conductor $456$
Sign $0.233 + 0.972i$
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  + (0.954 − 1.04i)2-s + (1.61 + 0.631i)3-s + (−0.178 − 1.99i)4-s + (1.99 − 3.45i)5-s + (2.19 − 1.08i)6-s + 0.938i·7-s + (−2.24 − 1.71i)8-s + (2.20 + 2.03i)9-s + (−1.70 − 5.37i)10-s + 4.53i·11-s + (0.969 − 3.32i)12-s + (−1.90 + 1.09i)13-s + (0.979 + 0.895i)14-s + (5.39 − 4.30i)15-s + (−3.93 + 0.710i)16-s + (−3.59 − 2.07i)17-s + ⋯
L(s)  = 1  + (0.674 − 0.737i)2-s + (0.931 + 0.364i)3-s + (−0.0891 − 0.996i)4-s + (0.891 − 1.54i)5-s + (0.897 − 0.441i)6-s + 0.354i·7-s + (−0.795 − 0.606i)8-s + (0.734 + 0.678i)9-s + (−0.537 − 1.69i)10-s + 1.36i·11-s + (0.279 − 0.960i)12-s + (−0.528 + 0.305i)13-s + (0.261 + 0.239i)14-s + (1.39 − 1.11i)15-s + (−0.984 + 0.177i)16-s + (−0.871 − 0.503i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.233 + 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.233 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(456\)    =    \(2^{3} \cdot 3 \cdot 19\)
Sign: $0.233 + 0.972i$
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: \((2,\ 456,\ (\ :1/2),\ 0.233 + 0.972i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.20993 - 1.74242i\)
\(L(\frac12)\) \(\approx\) \(2.20993 - 1.74242i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.954 + 1.04i)T \)
3 \( 1 + (-1.61 - 0.631i)T \)
19 \( 1 + (3.77 - 2.17i)T \)
good5 \( 1 + (-1.99 + 3.45i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 - 0.938iT - 7T^{2} \)
11 \( 1 - 4.53iT - 11T^{2} \)
13 \( 1 + (1.90 - 1.09i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.59 + 2.07i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (-2.04 - 3.53i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.79 - 4.83i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 3.09iT - 31T^{2} \)
37 \( 1 + 9.54iT - 37T^{2} \)
41 \( 1 + (-9.86 - 5.69i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.21 + 3.83i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.34 + 2.33i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.96 + 6.86i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.471 + 0.272i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (11.1 - 6.45i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.848 + 1.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.0740 + 0.128i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.99 - 3.46i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.47 + 3.73i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 1.80iT - 83T^{2} \)
89 \( 1 + (7.05 - 4.07i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.31 + 7.47i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.76492507783764701029459834899, −9.700289043282406232386217631805, −9.360608022530546038806644845333, −8.650335607553120111975220254501, −7.18362725016880405906026549327, −5.71597387494957657719131426507, −4.73319022675075588464478212259, −4.26185755730271827395063738486, −2.43646637885880606327206602292, −1.71385585925052350668919492411, 2.50513333842529706036224200244, 3.08688452810260363058717196292, 4.35756718489834198319515829880, 6.09052820326345037923913581559, 6.50542896976998602061671160897, 7.39167588392823650283854801422, 8.350598974790187222840820711280, 9.229358367976629077270453461432, 10.46411687749569101190266619769, 11.13476520861835757208045370796

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