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Properties

Label 2-462-11.4-c1-0-11
Degree $2$
Conductor $462$
Sign $0.859 + 0.511i$
Analytic cond. $3.68908$
Root an. cond. $1.92070$
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  + (0.809 + 0.587i)2-s + (0.309 − 0.951i)3-s + (0.309 + 0.951i)4-s + (2.92 − 2.12i)5-s + (0.809 − 0.587i)6-s + (−0.309 − 0.951i)7-s + (−0.309 + 0.951i)8-s + (−0.809 − 0.587i)9-s + 3.61·10-s + (0.809 − 3.21i)11-s + 12-s + (−2.61 − 1.90i)13-s + (0.309 − 0.951i)14-s + (−1.11 − 3.44i)15-s + (−0.809 + 0.587i)16-s + (−4.35 + 3.16i)17-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (0.178 − 0.549i)3-s + (0.154 + 0.475i)4-s + (1.30 − 0.951i)5-s + (0.330 − 0.239i)6-s + (−0.116 − 0.359i)7-s + (−0.109 + 0.336i)8-s + (−0.269 − 0.195i)9-s + 1.14·10-s + (0.243 − 0.969i)11-s + 0.288·12-s + (−0.726 − 0.527i)13-s + (0.0825 − 0.254i)14-s + (−0.288 − 0.888i)15-s + (−0.202 + 0.146i)16-s + (−1.05 + 0.767i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.859 + 0.511i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.859 + 0.511i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.25006 - 0.618848i\)
\(L(\frac12)\) \(\approx\) \(2.25006 - 0.618848i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 - 0.587i)T \)
3 \( 1 + (-0.309 + 0.951i)T \)
7 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (-0.809 + 3.21i)T \)
good5 \( 1 + (-2.92 + 2.12i)T + (1.54 - 4.75i)T^{2} \)
13 \( 1 + (2.61 + 1.90i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (4.35 - 3.16i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (2.19 - 6.74i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 6.32T + 23T^{2} \)
29 \( 1 + (-1 - 3.07i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-8.16 - 5.93i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (1.97 + 6.06i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.59 + 4.89i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 0.763T + 43T^{2} \)
47 \( 1 + (3.76 - 11.5i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-7.23 - 5.25i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-2.76 - 8.50i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-3.85 + 2.80i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 3.23T + 67T^{2} \)
71 \( 1 + (8.47 - 6.15i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (2.47 + 7.60i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (4.61 + 3.35i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (8.70 - 6.32i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 11.8T + 89T^{2} \)
97 \( 1 + (5.47 + 3.97i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.96053476947157183046759632488, −10.06426604220165115036243262276, −8.847227846805040385693065908896, −8.408268376367993875891875542723, −7.09070517651715862638362293703, −6.12100220900488345936825434537, −5.50528416350705273376961950473, −4.32578690140140855423390724882, −2.81975941803710514967787250089, −1.38746255571884355257263658326, 2.29681415650743962909510735681, 2.72144395257292170215943227090, 4.47084660319107994613558550203, 5.16929342702029011299789106407, 6.59141324374096121560048045181, 6.90824238562105365446727109219, 8.853149856570397140322786103131, 9.690168514880262019713811394662, 10.06134292860282091815183571856, 11.19386869447154749905347443275

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