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Properties

Label 2-462-21.20-c1-0-0
Degree $2$
Conductor $462$
Sign $0.387 - 0.921i$
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  − i·2-s + (−1.22 − 1.22i)3-s − 4-s − 2.44·5-s + (−1.22 + 1.22i)6-s + (−1 − 2.44i)7-s + i·8-s + 2.99i·9-s + 2.44i·10-s + i·11-s + (1.22 + 1.22i)12-s − 2.44i·13-s + (−2.44 + i)14-s + (2.99 + 2.99i)15-s + 16-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.707 − 0.707i)3-s − 0.5·4-s − 1.09·5-s + (−0.499 + 0.499i)6-s + (−0.377 − 0.925i)7-s + 0.353i·8-s + 0.999i·9-s + 0.774i·10-s + 0.301i·11-s + (0.353 + 0.353i)12-s − 0.679i·13-s + (−0.654 + 0.267i)14-s + (0.774 + 0.774i)15-s + 0.250·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.387 - 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.387 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.387 - 0.921i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.387 - 0.921i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0967315 + 0.0642775i\)
\(L(\frac12)\) \(\approx\) \(0.0967315 + 0.0642775i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (1.22 + 1.22i)T \)
7 \( 1 + (1 + 2.44i)T \)
11 \( 1 - iT \)
good5 \( 1 + 2.44T + 5T^{2} \)
13 \( 1 + 2.44iT - 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 7.34iT - 19T^{2} \)
23 \( 1 - 6iT - 23T^{2} \)
29 \( 1 + 6iT - 29T^{2} \)
31 \( 1 - 4.89iT - 31T^{2} \)
37 \( 1 - 10T + 37T^{2} \)
41 \( 1 + 9.79T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + 9.79T + 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 - 2.44T + 59T^{2} \)
61 \( 1 - 7.34iT - 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 - 4.89iT - 73T^{2} \)
79 \( 1 + 10T + 79T^{2} \)
83 \( 1 - 2.44T + 83T^{2} \)
89 \( 1 + 9.79T + 89T^{2} \)
97 \( 1 + 4.89iT - 97T^{2} \)
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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

−11.57990848773864484201430917374, −10.37576878615702557856753007480, −9.898635745113005659711882496898, −8.112706003773901539956623065434, −7.76759328553769945353209766186, −6.67987606790364333254638707177, −5.47112837281946220296382120675, −4.24453764565200659267453354999, −3.31582103246861352312865552201, −1.43698038223974440069218996679, 0.083167276507984440073228032619, 3.11455631462562314526786438577, 4.34708332995483595406922350632, 5.07780813897583773189660736273, 6.29079906958839305790845835259, 6.93134346106320501603740350400, 8.294936047873549683168138768354, 8.994816712616686990009287475782, 9.821893398304786511969182272119, 11.10653484227962410446917969107

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