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Properties

Label 2-462-7.2-c1-0-0
Degree $2$
Conductor $462$
Sign $0.605 - 0.795i$
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  + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (−1.5 + 2.59i)5-s − 0.999·6-s + (−2.5 + 0.866i)7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (1.5 + 2.59i)10-s + (0.5 + 0.866i)11-s + (−0.499 + 0.866i)12-s + 6·13-s + (−0.500 + 2.59i)14-s + 3·15-s + (−0.5 + 0.866i)16-s + (2.5 + 4.33i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.670 + 1.16i)5-s − 0.408·6-s + (−0.944 + 0.327i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (0.474 + 0.821i)10-s + (0.150 + 0.261i)11-s + (−0.144 + 0.249i)12-s + 1.66·13-s + (−0.133 + 0.694i)14-s + 0.774·15-s + (−0.125 + 0.216i)16-s + (0.606 + 1.05i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.605 - 0.795i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.819390 + 0.406194i\)
\(L(\frac12)\) \(\approx\) \(0.819390 + 0.406194i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (2.5 - 0.866i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (1.5 - 2.59i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 - 6T + 13T^{2} \)
17 \( 1 + (-2.5 - 4.33i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3 - 5.19i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.5 - 4.33i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1 + 1.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 5T + 41T^{2} \)
43 \( 1 + 10T + 43T^{2} \)
47 \( 1 + (4.5 - 7.79i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1 + 1.73i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6 - 10.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.5 + 4.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.5 + 4.33i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 4T + 71T^{2} \)
73 \( 1 + (6 + 10.3i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - T + 83T^{2} \)
89 \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 9T + 97T^{2} \)
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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

−11.20563809397101014650664970573, −10.57563800529633923998914339475, −9.677191152823600531507542156007, −8.401261182946546531249815981879, −7.44274598769626041223788770633, −6.19563026165788213932908796184, −5.93771831703104399924578753623, −3.82307835903289348310255895215, −3.41053038581358369306414975653, −1.80139740433306722210677912276, 0.53287437485740683105990951494, 3.36858210766550814086785619680, 4.17754879571862179316054462549, 5.13683434004154946567090740583, 6.17017495147869987393925431088, 7.07203850182775116138844693569, 8.415653659651287878995620351852, 8.866691796209111222041956602010, 9.872543175763159238176105725147, 11.08596382079880848787592403646

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