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Properties

Label 2-456-24.11-c1-0-32
Degree $2$
Conductor $456$
Sign $0.691 + 0.722i$
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  + (−0.366 − 1.36i)2-s + (1.73 + 0.0380i)3-s + (−1.73 + 1.00i)4-s + 1.96·5-s + (−0.582 − 2.37i)6-s + 1.36i·7-s + (2.00 + 1.99i)8-s + (2.99 + 0.131i)9-s + (−0.718 − 2.68i)10-s − 3.24i·11-s + (−3.03 + 1.66i)12-s + 6.13i·13-s + (1.86 − 0.501i)14-s + (3.39 + 0.0746i)15-s + (1.99 − 3.46i)16-s − 1.59i·17-s + ⋯
L(s)  = 1  + (−0.259 − 0.965i)2-s + (0.999 + 0.0219i)3-s + (−0.865 + 0.500i)4-s + 0.877·5-s + (−0.237 − 0.971i)6-s + 0.516i·7-s + (0.707 + 0.706i)8-s + (0.999 + 0.0439i)9-s + (−0.227 − 0.847i)10-s − 0.978i·11-s + (−0.876 + 0.481i)12-s + 1.70i·13-s + (0.499 − 0.133i)14-s + (0.877 + 0.0192i)15-s + (0.499 − 0.866i)16-s − 0.385i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.691 + 0.722i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.691 + 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(456\)    =    \(2^{3} \cdot 3 \cdot 19\)
Sign: $0.691 + 0.722i$
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: \((2,\ 456,\ (\ :1/2),\ 0.691 + 0.722i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.70011 - 0.725557i\)
\(L(\frac12)\) \(\approx\) \(1.70011 - 0.725557i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.366 + 1.36i)T \)
3 \( 1 + (-1.73 - 0.0380i)T \)
19 \( 1 - T \)
good5 \( 1 - 1.96T + 5T^{2} \)
7 \( 1 - 1.36iT - 7T^{2} \)
11 \( 1 + 3.24iT - 11T^{2} \)
13 \( 1 - 6.13iT - 13T^{2} \)
17 \( 1 + 1.59iT - 17T^{2} \)
23 \( 1 - 3.57T + 23T^{2} \)
29 \( 1 + 2.53T + 29T^{2} \)
31 \( 1 + 6.19iT - 31T^{2} \)
37 \( 1 + 6.07iT - 37T^{2} \)
41 \( 1 - 5.65iT - 41T^{2} \)
43 \( 1 - 2.89T + 43T^{2} \)
47 \( 1 + 13.2T + 47T^{2} \)
53 \( 1 + 12.0T + 53T^{2} \)
59 \( 1 + 6.55iT - 59T^{2} \)
61 \( 1 + 2.43iT - 61T^{2} \)
67 \( 1 + 6.04T + 67T^{2} \)
71 \( 1 - 11.2T + 71T^{2} \)
73 \( 1 + 13.0T + 73T^{2} \)
79 \( 1 - 2.08iT - 79T^{2} \)
83 \( 1 + 10.0iT - 83T^{2} \)
89 \( 1 - 15.8iT - 89T^{2} \)
97 \( 1 - 4.59T + 97T^{2} \)
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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

−11.00678717035788801923664543218, −9.669962253774770087867576421487, −9.374213658595223644165890688225, −8.653758081766023818146080783501, −7.61442480107614953359610904763, −6.27657982805371054989780618974, −4.90686336509128715577647662739, −3.71840445196144485514234234166, −2.58694188067039694876447460780, −1.66018493694154790283853453735, 1.50596490069147911627395076151, 3.20600416442081693025584524436, 4.58449960851281086372343911814, 5.58611350350943640806499230964, 6.79136943996307495374494186671, 7.57152690126272918119399767052, 8.344239005675977680943823903258, 9.339968097436659850136765012754, 10.05158067955216591849792644947, 10.55996600642808238018648562895

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