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Properties

Label 2-462-33.2-c1-0-19
Degree $2$
Conductor $462$
Sign $0.111 + 0.993i$
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  + (−0.309 + 0.951i)2-s + (1.31 − 1.13i)3-s + (−0.809 − 0.587i)4-s + (−3.37 + 1.09i)5-s + (0.670 + 1.59i)6-s + (0.587 − 0.809i)7-s + (0.809 − 0.587i)8-s + (0.442 − 2.96i)9-s − 3.54i·10-s + (1.81 − 2.77i)11-s + (−1.72 + 0.143i)12-s + (−5.58 − 1.81i)13-s + (0.587 + 0.809i)14-s + (−3.18 + 5.25i)15-s + (0.309 + 0.951i)16-s + (−2.31 − 7.12i)17-s + ⋯
L(s)  = 1  + (−0.218 + 0.672i)2-s + (0.757 − 0.652i)3-s + (−0.404 − 0.293i)4-s + (−1.50 + 0.490i)5-s + (0.273 + 0.652i)6-s + (0.222 − 0.305i)7-s + (0.286 − 0.207i)8-s + (0.147 − 0.989i)9-s − 1.12i·10-s + (0.546 − 0.837i)11-s + (−0.498 + 0.0414i)12-s + (−1.54 − 0.503i)13-s + (0.157 + 0.216i)14-s + (−0.822 + 1.35i)15-s + (0.0772 + 0.237i)16-s + (−0.561 − 1.72i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.111 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.111 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.111 + 0.993i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (365, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.111 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.646956 - 0.578625i\)
\(L(\frac12)\) \(\approx\) \(0.646956 - 0.578625i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.309 - 0.951i)T \)
3 \( 1 + (-1.31 + 1.13i)T \)
7 \( 1 + (-0.587 + 0.809i)T \)
11 \( 1 + (-1.81 + 2.77i)T \)
good5 \( 1 + (3.37 - 1.09i)T + (4.04 - 2.93i)T^{2} \)
13 \( 1 + (5.58 + 1.81i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (2.31 + 7.12i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-2.87 - 3.95i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 + 3.38iT - 23T^{2} \)
29 \( 1 + (-4.51 - 3.27i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (0.123 - 0.381i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (8.42 + 6.11i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-0.577 + 0.419i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 0.722iT - 43T^{2} \)
47 \( 1 + (0.656 + 0.903i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-0.334 - 0.108i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (0.843 - 1.16i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (1.21 - 0.393i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 - 9.40T + 67T^{2} \)
71 \( 1 + (3.46 - 1.12i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (1.72 - 2.38i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-11.2 - 3.65i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (-3.11 - 9.57i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 4.97iT - 89T^{2} \)
97 \( 1 + (0.155 - 0.479i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.87825016947481407748988271711, −9.701192482082172088328363840293, −8.708445608033499826228620765900, −7.917506869038230087252947234388, −7.26875117301682761491801056712, −6.74800843135644896404811928492, −5.08557449236519902898502304638, −3.84740279196797815267507519505, −2.82574228306179939395378500308, −0.52879682928014690463598021048, 1.98018415811270412193463699995, 3.38017383128367315429948519740, 4.39048932034106303703779799899, 4.85191564976795247475912138115, 7.07816540567268755309267644524, 7.917582724661326939328760174681, 8.665845695687814896412779492769, 9.430367799673521141370669547432, 10.27437895412673795009257299966, 11.35905341583147790948268941517

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