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Properties

Label 2-420-12.11-c1-0-3
Degree $2$
Conductor $420$
Sign $0.349 - 0.936i$
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  + (−0.859 − 1.12i)2-s + (−1.72 − 0.161i)3-s + (−0.521 + 1.93i)4-s − i·5-s + (1.30 + 2.07i)6-s − i·7-s + (2.61 − 1.07i)8-s + (2.94 + 0.556i)9-s + (−1.12 + 0.859i)10-s − 5.30·11-s + (1.21 − 3.24i)12-s − 3.33·13-s + (−1.12 + 0.859i)14-s + (−0.161 + 1.72i)15-s + (−3.45 − 2.01i)16-s + 0.634i·17-s + ⋯
L(s)  = 1  + (−0.607 − 0.793i)2-s + (−0.995 − 0.0931i)3-s + (−0.260 + 0.965i)4-s − 0.447i·5-s + (0.531 + 0.847i)6-s − 0.377i·7-s + (0.924 − 0.379i)8-s + (0.982 + 0.185i)9-s + (−0.355 + 0.271i)10-s − 1.59·11-s + (0.349 − 0.936i)12-s − 0.924·13-s + (−0.300 + 0.229i)14-s + (−0.0416 + 0.445i)15-s + (−0.864 − 0.503i)16-s + 0.154i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.349 - 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.349 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.349 - 0.936i$
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: \((2,\ 420,\ (\ :1/2),\ 0.349 - 0.936i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.209744 + 0.145615i\)
\(L(\frac12)\) \(\approx\) \(0.209744 + 0.145615i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.859 + 1.12i)T \)
3 \( 1 + (1.72 + 0.161i)T \)
5 \( 1 + iT \)
7 \( 1 + iT \)
good11 \( 1 + 5.30T + 11T^{2} \)
13 \( 1 + 3.33T + 13T^{2} \)
17 \( 1 - 0.634iT - 17T^{2} \)
19 \( 1 - 5.08iT - 19T^{2} \)
23 \( 1 - 8.45T + 23T^{2} \)
29 \( 1 - 10.4iT - 29T^{2} \)
31 \( 1 - 4.35iT - 31T^{2} \)
37 \( 1 + 0.255T + 37T^{2} \)
41 \( 1 - 5.20iT - 41T^{2} \)
43 \( 1 + 6.21iT - 43T^{2} \)
47 \( 1 + 13.0T + 47T^{2} \)
53 \( 1 - 8.71iT - 53T^{2} \)
59 \( 1 - 1.82T + 59T^{2} \)
61 \( 1 + 1.58T + 61T^{2} \)
67 \( 1 - 9.38iT - 67T^{2} \)
71 \( 1 + 2.69T + 71T^{2} \)
73 \( 1 + 9.27T + 73T^{2} \)
79 \( 1 + 5.23iT - 79T^{2} \)
83 \( 1 - 4.47T + 83T^{2} \)
89 \( 1 + 11.3iT - 89T^{2} \)
97 \( 1 + 10.9T + 97T^{2} \)
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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

−11.18214887332995422097765589065, −10.48078459058260499906590714897, −9.961144045318693176178813841295, −8.746437973043613894521578646339, −7.68151347646304350894257672618, −6.99568914299182280047675297865, −5.32005611201553121256095359446, −4.67685232259200396767110404358, −3.09878435053208023955198410602, −1.43200471192372019272126834408, 0.23322306504050473064795606896, 2.49409049064980357873748055737, 4.75790813276313078912476810636, 5.30319540885173823663485438344, 6.39541635425794386377679459254, 7.24970833884877134249071553459, 8.010657138207129636866926136310, 9.401257944805131942041292924447, 10.03181844330137394570823888888, 10.94676853084745181898049749087

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