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Properties

Label 2-420-12.11-c1-0-35
Degree $2$
Conductor $420$
Sign $0.669 + 0.742i$
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 − 0.757i)2-s + (−0.667 + 1.59i)3-s + (0.853 − 1.80i)4-s − i·5-s + (0.412 + 2.41i)6-s − i·7-s + (−0.350 − 2.80i)8-s + (−2.10 − 2.13i)9-s + (−0.757 − 1.19i)10-s + 6.52·11-s + (2.32 + 2.57i)12-s + 0.852·13-s + (−0.757 − 1.19i)14-s + (1.59 + 0.667i)15-s + (−2.54 − 3.08i)16-s − 4.75i·17-s + ⋯
L(s)  = 1  + (0.844 − 0.535i)2-s + (−0.385 + 0.922i)3-s + (0.426 − 0.904i)4-s − 0.447i·5-s + (0.168 + 0.985i)6-s − 0.377i·7-s + (−0.123 − 0.992i)8-s + (−0.702 − 0.711i)9-s + (−0.239 − 0.377i)10-s + 1.96·11-s + (0.669 + 0.742i)12-s + 0.236·13-s + (−0.202 − 0.319i)14-s + (0.412 + 0.172i)15-s + (−0.635 − 0.771i)16-s − 1.15i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.669 + 0.742i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.669 + 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.669 + 0.742i$
Analytic conductor: \(3.35371\)
Root analytic conductor: \(1.83131\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{420} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 420,\ (\ :1/2),\ 0.669 + 0.742i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.87505 - 0.833563i\)
\(L(\frac12)\) \(\approx\) \(1.87505 - 0.833563i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.19 + 0.757i)T \)
3 \( 1 + (0.667 - 1.59i)T \)
5 \( 1 + iT \)
7 \( 1 + iT \)
good11 \( 1 - 6.52T + 11T^{2} \)
13 \( 1 - 0.852T + 13T^{2} \)
17 \( 1 + 4.75iT - 17T^{2} \)
19 \( 1 - 5.33iT - 19T^{2} \)
23 \( 1 - 0.680T + 23T^{2} \)
29 \( 1 + 0.963iT - 29T^{2} \)
31 \( 1 - 7.17iT - 31T^{2} \)
37 \( 1 + 1.92T + 37T^{2} \)
41 \( 1 - 6.55iT - 41T^{2} \)
43 \( 1 + 8.66iT - 43T^{2} \)
47 \( 1 + 8.50T + 47T^{2} \)
53 \( 1 - 1.34iT - 53T^{2} \)
59 \( 1 - 0.990T + 59T^{2} \)
61 \( 1 + 9.34T + 61T^{2} \)
67 \( 1 - 12.1iT - 67T^{2} \)
71 \( 1 + 6.95T + 71T^{2} \)
73 \( 1 - 9.97T + 73T^{2} \)
79 \( 1 - 12.5iT - 79T^{2} \)
83 \( 1 + 9.19T + 83T^{2} \)
89 \( 1 + 1.22iT - 89T^{2} \)
97 \( 1 - 11.1T + 97T^{2} \)
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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

−11.32525681767233412684274361185, −10.23734321242000767220116008719, −9.551740137868093463256892422779, −8.730838984132647057379120141263, −6.93665712422456567695878796686, −6.07474739541689662229986839282, −5.01582529666650640719317081729, −4.12304101079086199677951278094, −3.35742286598429701133060041109, −1.27108743305962113384411999979, 1.86882680895337157370337052289, 3.33501351073675698993249343111, 4.57517242105529760115018171772, 5.99111176488410676064005170334, 6.43341695354216217865166303736, 7.25376940943102682624912411771, 8.327476279319595775300364225006, 9.220375934680462155664485785676, 10.96258579504380934761695318093, 11.53545176961184505142486622061

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