Properties

Label 2-420-60.59-c1-0-47
Degree $2$
Conductor $420$
Sign $0.367 + 0.930i$
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  + (−0.621 − 1.27i)2-s + (1.72 + 0.121i)3-s + (−1.22 + 1.57i)4-s + (1 − 2i)5-s + (−0.919 − 2.27i)6-s + 7-s + (2.76 + 0.578i)8-s + (2.97 + 0.419i)9-s + (−3.16 − 0.0276i)10-s + 3.45·11-s + (−2.31 + 2.57i)12-s + 4.83i·13-s + (−0.621 − 1.27i)14-s + (1.97 − 3.33i)15-s + (−0.985 − 3.87i)16-s − 5.94·17-s + ⋯
L(s)  = 1  + (−0.439 − 0.898i)2-s + (0.997 + 0.0700i)3-s + (−0.613 + 0.789i)4-s + (0.447 − 0.894i)5-s + (−0.375 − 0.926i)6-s + 0.377·7-s + (0.978 + 0.204i)8-s + (0.990 + 0.139i)9-s + (−0.999 − 0.00874i)10-s + 1.04·11-s + (−0.667 + 0.744i)12-s + 1.34i·13-s + (−0.166 − 0.339i)14-s + (0.508 − 0.860i)15-s + (−0.246 − 0.969i)16-s − 1.44·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.367 + 0.930i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.367 + 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.367 + 0.930i$
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: \((2,\ 420,\ (\ :1/2),\ 0.367 + 0.930i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39004 - 0.945651i\)
\(L(\frac12)\) \(\approx\) \(1.39004 - 0.945651i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.621 + 1.27i)T \)
3 \( 1 + (-1.72 - 0.121i)T \)
5 \( 1 + (-1 + 2i)T \)
7 \( 1 - T \)
good11 \( 1 - 3.45T + 11T^{2} \)
13 \( 1 - 4.83iT - 13T^{2} \)
17 \( 1 + 5.94T + 17T^{2} \)
19 \( 1 + 1.08iT - 19T^{2} \)
23 \( 1 - 0.596iT - 23T^{2} \)
29 \( 1 + 4.83iT - 29T^{2} \)
31 \( 1 + 9.56iT - 31T^{2} \)
37 \( 1 - 2.91iT - 37T^{2} \)
41 \( 1 - 6.91iT - 41T^{2} \)
43 \( 1 - 7.39T + 43T^{2} \)
47 \( 1 + 0.242iT - 47T^{2} \)
53 \( 1 + 11.8T + 53T^{2} \)
59 \( 1 - 3.25T + 59T^{2} \)
61 \( 1 + 12.6T + 61T^{2} \)
67 \( 1 + 5.71T + 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 - 8iT - 73T^{2} \)
79 \( 1 - 0.949iT - 79T^{2} \)
83 \( 1 - 16.9iT - 83T^{2} \)
89 \( 1 - 5.23iT - 89T^{2} \)
97 \( 1 + 3.30iT - 97T^{2} \)
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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

−11.12032017962887595124076607093, −9.688149118156443203728150855657, −9.297179360771287514173378340793, −8.671113014564968156702217359399, −7.75015849915149071157319870093, −6.47498175631935730780652735514, −4.47966605557148339590012881237, −4.18175171089377403794672115766, −2.39756529951137475425605025371, −1.48702576703692628709898045689, 1.70651328880567396191403880815, 3.25991482964941801418624682659, 4.59884925978968579407595531668, 5.98655374117242715846154090888, 6.91602830983979071661029809374, 7.59237888891814503443928691959, 8.714604647195644652880670203631, 9.208707060766649844772330213310, 10.37647187539469739660922524985, 10.84532164760799796450634542704

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