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Properties

Label 2-462-77.13-c1-0-14
Degree $2$
Conductor $462$
Sign $0.926 - 0.376i$
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  + (0.951 + 0.309i)2-s + (0.587 + 0.809i)3-s + (0.809 + 0.587i)4-s + (3.18 − 1.03i)5-s + (0.309 + 0.951i)6-s + (−1.29 − 2.30i)7-s + (0.587 + 0.809i)8-s + (−0.309 + 0.951i)9-s + 3.35·10-s + (0.695 + 3.24i)11-s + 0.999i·12-s + (1.25 − 3.86i)13-s + (−0.516 − 2.59i)14-s + (2.71 + 1.96i)15-s + (0.309 + 0.951i)16-s + (−1.34 − 4.13i)17-s + ⋯
L(s)  = 1  + (0.672 + 0.218i)2-s + (0.339 + 0.467i)3-s + (0.404 + 0.293i)4-s + (1.42 − 0.463i)5-s + (0.126 + 0.388i)6-s + (−0.488 − 0.872i)7-s + (0.207 + 0.286i)8-s + (−0.103 + 0.317i)9-s + 1.05·10-s + (0.209 + 0.977i)11-s + 0.288i·12-s + (0.348 − 1.07i)13-s + (−0.138 − 0.693i)14-s + (0.699 + 0.508i)15-s + (0.0772 + 0.237i)16-s + (−0.325 − 1.00i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 - 0.376i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.926 - 0.376i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.926 - 0.376i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.926 - 0.376i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.61325 + 0.510639i\)
\(L(\frac12)\) \(\approx\) \(2.61325 + 0.510639i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.951 - 0.309i)T \)
3 \( 1 + (-0.587 - 0.809i)T \)
7 \( 1 + (1.29 + 2.30i)T \)
11 \( 1 + (-0.695 - 3.24i)T \)
good5 \( 1 + (-3.18 + 1.03i)T + (4.04 - 2.93i)T^{2} \)
13 \( 1 + (-1.25 + 3.86i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (1.34 + 4.13i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (6.45 - 4.68i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 1.33T + 23T^{2} \)
29 \( 1 + (5.08 - 6.99i)T + (-8.96 - 27.5i)T^{2} \)
31 \( 1 + (-4.15 - 1.35i)T + (25.0 + 18.2i)T^{2} \)
37 \( 1 + (4.73 + 3.44i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (3.41 - 2.48i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 7.53iT - 43T^{2} \)
47 \( 1 + (-1.67 - 2.31i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-4.02 + 12.3i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-1.54 + 2.11i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (-2.11 - 6.49i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 1.42T + 67T^{2} \)
71 \( 1 + (3.40 + 10.4i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (6.22 + 4.52i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-8.00 - 2.60i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (5.05 + 15.5i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 8.96iT - 89T^{2} \)
97 \( 1 + (-3.40 - 1.10i)T + (78.4 + 57.0i)T^{2} \)
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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

−10.82176009922165380880651088761, −10.18663172254181409573055613214, −9.487686708616358657437592060834, −8.467513956564352768786312765128, −7.22952083990520150275694525906, −6.30838412207896275125897910653, −5.31506073520030778644031108811, −4.42572508656490360077350462744, −3.21218730089242318353327023750, −1.82212371317594982362057824118, 1.93185834493370342266768353747, 2.62471769920339475445800937476, 3.99118717672593199351240010983, 5.60258634799951154325337204324, 6.30082519642492866316647239590, 6.75388376161705735192787041769, 8.597601122629639290748718513782, 9.102732291267151056868651694363, 10.18870216266478462493549519805, 11.08051420444941022212027558421

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