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Properties

Label 2-456-456.11-c1-0-3
Degree $2$
Conductor $456$
Sign $0.658 - 0.752i$
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  + (0.707 − 1.22i)2-s + (−1.18 − 1.26i)3-s + (−0.999 − 1.73i)4-s + (−0.707 + 1.22i)5-s + (−2.38 + 0.560i)6-s + 4.69i·7-s − 2.82·8-s + (−0.186 + 2.99i)9-s + (0.999 + 1.73i)10-s + 3.31i·11-s + (−1.00 + 3.31i)12-s + (5.74 + 3.31i)14-s + (2.38 − 0.560i)15-s + (−2.00 + 3.46i)16-s + (−5.74 − 3.31i)17-s + (3.53 + 2.34i)18-s + ⋯
L(s)  = 1  + (0.499 − 0.866i)2-s + (−0.684 − 0.728i)3-s + (−0.499 − 0.866i)4-s + (−0.316 + 0.547i)5-s + (−0.973 + 0.228i)6-s + 1.77i·7-s − 0.999·8-s + (−0.0620 + 0.998i)9-s + (0.316 + 0.547i)10-s + 1.00i·11-s + (−0.288 + 0.957i)12-s + (1.53 + 0.886i)14-s + (0.615 − 0.144i)15-s + (−0.500 + 0.866i)16-s + (−1.39 − 0.804i)17-s + (0.833 + 0.552i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.658 - 0.752i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.658 - 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(456\)    =    \(2^{3} \cdot 3 \cdot 19\)
Sign: $0.658 - 0.752i$
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: \((2,\ 456,\ (\ :1/2),\ 0.658 - 0.752i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.627810 + 0.284986i\)
\(L(\frac12)\) \(\approx\) \(0.627810 + 0.284986i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 1.22i)T \)
3 \( 1 + (1.18 + 1.26i)T \)
19 \( 1 + (3.5 + 2.59i)T \)
good5 \( 1 + (0.707 - 1.22i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 - 4.69iT - 7T^{2} \)
11 \( 1 - 3.31iT - 11T^{2} \)
13 \( 1 + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (5.74 + 3.31i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (-2.12 - 3.67i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.53 + 6.12i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.69iT - 31T^{2} \)
37 \( 1 + 4.69iT - 37T^{2} \)
41 \( 1 + (2.87 + 1.65i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (3 - 5.19i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.53 - 6.12i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.65 - 9.79i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.87 - 1.65i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.06 + 2.34i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.5 + 4.33i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.82 - 4.89i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (0.5 - 0.866i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 3.31iT - 83T^{2} \)
89 \( 1 + (-5.74 + 3.31i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.5 + 12.9i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.43944349353837923232527327230, −10.74389618586341222727891276932, −9.444364936305717170248128462593, −8.750324944179945114071721598131, −7.26472636666853823853965405915, −6.36089949610121010246194270003, −5.40713648178375429173378392527, −4.55247499709078532007472242008, −2.73048603054770891669011116634, −2.01434817500569153461689140044, 0.38324534099407349936856314000, 3.67305849385407014594192954481, 4.19099473130037243587070930841, 5.07583137869169428947711853338, 6.33494710169943394268041407607, 6.93013444860750752822114609189, 8.275202630065367374037302772099, 8.851677041723046909954043276291, 10.27001962119703499691475388296, 10.87105195401471995605516960786

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