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Properties

Label 2-420-60.59-c1-0-5
Degree $2$
Conductor $420$
Sign $-0.897 + 0.441i$
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 + (−0.856 − 2.29i)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 + (2.75 + 2.09i)12-s + 6.41i·13-s + (−1.19 + 0.756i)14-s + (−3.03 − 2.40i)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 + (−0.349 − 0.936i)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 + (0.796 + 0.604i)12-s + 1.77i·13-s + (−0.319 + 0.202i)14-s + (−0.783 − 0.621i)15-s + (−0.633 − 0.773i)16-s − 1.33·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.897 + 0.441i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.897 + 0.441i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.897 + 0.441i$
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: \((2,\ 420,\ (\ :1/2),\ -0.897 + 0.441i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.113377 - 0.486814i\)
\(L(\frac12)\) \(\approx\) \(0.113377 - 0.486814i\)
\(L(\frac{3}{2})\) 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

−11.10469833079173376170621882573, −10.96002810041682563957061289221, −9.827766459743483151661086542403, −9.003117277565327167129180755038, −8.280558575170152704044185077697, −6.85159786632118254473083289304, −6.46957379247151135507848104662, −4.96391956171923386843744321343, −3.99125724867813844576208841508, −2.27356208387525985556943881837, 0.42301038969498999891358237875, 1.75407393188271888618815557873, 3.22013043251160656977204830163, 4.79243214934287415579099173264, 6.08034117515336044110464653434, 7.35295336245163607184409396576, 8.051936239733962736996460721913, 8.624089668544385541744295025493, 9.700275898015510648445572314219, 10.93021540624688511664868935016

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