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Properties

Label 2-420-60.59-c1-0-28
Degree $2$
Conductor $420$
Sign $0.252 - 0.967i$
Analytic cond. $3.35371$
Root an. cond. $1.83131$
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  + (−0.538 + 1.30i)2-s + (−0.520 + 1.65i)3-s + (−1.42 − 1.40i)4-s + (1.50 − 1.65i)5-s + (−1.88 − 1.56i)6-s + 7-s + (2.60 − 1.10i)8-s + (−2.45 − 1.72i)9-s + (1.34 + 2.85i)10-s + 5.78·11-s + (3.06 − 1.61i)12-s − 1.25i·13-s + (−0.538 + 1.30i)14-s + (1.94 + 3.34i)15-s + (0.0371 + 3.99i)16-s + 2.15·17-s + ⋯
L(s)  = 1  + (−0.380 + 0.924i)2-s + (−0.300 + 0.953i)3-s + (−0.710 − 0.703i)4-s + (0.673 − 0.738i)5-s + (−0.767 − 0.640i)6-s + 0.377·7-s + (0.921 − 0.389i)8-s + (−0.819 − 0.573i)9-s + (0.426 + 0.904i)10-s + 1.74·11-s + (0.884 − 0.465i)12-s − 0.347i·13-s + (−0.143 + 0.349i)14-s + (0.502 + 0.864i)15-s + (0.00928 + 0.999i)16-s + 0.522·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.252 - 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.252 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.252 - 0.967i$
Analytic conductor: \(3.35371\)
Root analytic conductor: \(1.83131\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{420} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 420,\ (\ :1/2),\ 0.252 - 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.952307 + 0.736033i\)
\(L(\frac12)\) \(\approx\) \(0.952307 + 0.736033i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.538 - 1.30i)T \)
3 \( 1 + (0.520 - 1.65i)T \)
5 \( 1 + (-1.50 + 1.65i)T \)
7 \( 1 - T \)
good11 \( 1 - 5.78T + 11T^{2} \)
13 \( 1 + 1.25iT - 13T^{2} \)
17 \( 1 - 2.15T + 17T^{2} \)
19 \( 1 - 1.25iT - 19T^{2} \)
23 \( 1 - 6.02iT - 23T^{2} \)
29 \( 1 + 3.22iT - 29T^{2} \)
31 \( 1 + 6.00iT - 31T^{2} \)
37 \( 1 - 5.63iT - 37T^{2} \)
41 \( 1 - 4.40iT - 41T^{2} \)
43 \( 1 + 6.28T + 43T^{2} \)
47 \( 1 - 8.45iT - 47T^{2} \)
53 \( 1 - 9.92T + 53T^{2} \)
59 \( 1 + 7.14T + 59T^{2} \)
61 \( 1 - 3.72T + 61T^{2} \)
67 \( 1 + 6.51T + 67T^{2} \)
71 \( 1 - 16.2T + 71T^{2} \)
73 \( 1 + 10.3iT - 73T^{2} \)
79 \( 1 - 7.68iT - 79T^{2} \)
83 \( 1 - 2.78iT - 83T^{2} \)
89 \( 1 + 17.0iT - 89T^{2} \)
97 \( 1 + 5.43iT - 97T^{2} \)
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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

−11.28163448188467148927115440510, −9.984980650218117979023070622659, −9.562257443520952843298168370155, −8.813112930212620725414231341919, −7.88250060596714064457986210798, −6.40228375386940725207865250807, −5.73888636617767923313700309993, −4.79245534878804854079399606168, −3.84912690000781760257024232087, −1.26736306664431985801856401492, 1.29604490932082026252423589324, 2.33659760588008146877111846257, 3.69821286941310666465751605431, 5.21788785293235143003900533526, 6.56305431139073010188078823385, 7.16251232768624813929827471334, 8.497509667995239562238626329716, 9.182851443535585270183067812887, 10.34105123421129724340930830841, 11.06388930338551106850266155094

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