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Properties

Label 2-420-60.59-c1-0-26
Degree $2$
Conductor $420$
Sign $-0.643 - 0.765i$
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  + (1.19 + 0.756i)2-s + (−0.356 + 1.69i)3-s + (0.856 + 1.80i)4-s + (1 + 2i)5-s + (−1.70 + 1.75i)6-s + 7-s + (−0.343 + 2.80i)8-s + (−2.74 − 1.20i)9-s + (−0.317 + 3.14i)10-s − 0.712·11-s + (−3.36 + 0.807i)12-s − 6.41i·13-s + (1.19 + 0.756i)14-s + (−3.74 + 0.982i)15-s + (−2.53 + 3.09i)16-s + 5.49·17-s + ⋯
L(s)  = 1  + (0.845 + 0.534i)2-s + (−0.205 + 0.978i)3-s + (0.428 + 0.903i)4-s + (0.447 + 0.894i)5-s + (−0.697 + 0.716i)6-s + 0.377·7-s + (−0.121 + 0.992i)8-s + (−0.915 − 0.402i)9-s + (−0.100 + 0.994i)10-s − 0.214·11-s + (−0.972 + 0.233i)12-s − 1.77i·13-s + (0.319 + 0.202i)14-s + (−0.967 + 0.253i)15-s + (−0.633 + 0.773i)16-s + 1.33·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.643 - 0.765i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.643 - 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.643 - 0.765i$
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: \((2,\ 420,\ (\ :1/2),\ -0.643 - 0.765i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.932053 + 2.00082i\)
\(L(\frac12)\) \(\approx\) \(0.932053 + 2.00082i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.19 - 0.756i)T \)
3 \( 1 + (0.356 - 1.69i)T \)
5 \( 1 + (-1 - 2i)T \)
7 \( 1 - T \)
good11 \( 1 + 0.712T + 11T^{2} \)
13 \( 1 + 6.41iT - 13T^{2} \)
17 \( 1 - 5.49T + 17T^{2} \)
19 \( 1 + 0.975iT - 19T^{2} \)
23 \( 1 + 5.80iT - 23T^{2} \)
29 \( 1 - 6.41iT - 29T^{2} \)
31 \( 1 - 0.244iT - 31T^{2} \)
37 \( 1 - 5.42iT - 37T^{2} \)
41 \( 1 - 1.42iT - 41T^{2} \)
43 \( 1 + 8.20T + 43T^{2} \)
47 \( 1 + 3.39iT - 47T^{2} \)
53 \( 1 - 4.84T + 53T^{2} \)
59 \( 1 - 7.47T + 59T^{2} \)
61 \( 1 + 1.26T + 61T^{2} \)
67 \( 1 - 13.0T + 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + 8iT - 73T^{2} \)
79 \( 1 + 15.0iT - 79T^{2} \)
83 \( 1 - 7.96iT - 83T^{2} \)
89 \( 1 - 6.25iT - 89T^{2} \)
97 \( 1 + 18.0iT - 97T^{2} \)
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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

−11.48238261493298460064913825685, −10.52750677743530502452529543971, −10.12093898734672117851425839210, −8.598202121067710102108021931087, −7.76359238192588801030951559394, −6.58007523902451993987259154693, −5.56853513074128819412867263754, −5.01224628235022301811507059934, −3.50735196063074703782789855174, −2.82067696279039982380733581989, 1.29100392147110390733562118322, 2.20769190707857523623744196603, 3.94751327307647830301103170862, 5.18173113297807740402360381590, 5.84910047408575402630550357541, 6.92360441214418520343453436205, 7.994134064617705568239967389256, 9.195291664267133955317753124460, 10.03507637367814503021529961044, 11.43117207438538891404704708852

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