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Properties

Label 2-420-12.11-c1-0-17
Degree $2$
Conductor $420$
Sign $0.984 - 0.173i$
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  + (−1.41 − 0.0678i)2-s + (1.66 − 0.463i)3-s + (1.99 + 0.191i)4-s + i·5-s + (−2.38 + 0.540i)6-s + i·7-s + (−2.79 − 0.405i)8-s + (2.57 − 1.54i)9-s + (0.0678 − 1.41i)10-s − 0.648·11-s + (3.41 − 0.601i)12-s + 4.96·13-s + (0.0678 − 1.41i)14-s + (0.463 + 1.66i)15-s + (3.92 + 0.763i)16-s + 6.88i·17-s + ⋯
L(s)  = 1  + (−0.998 − 0.0479i)2-s + (0.963 − 0.267i)3-s + (0.995 + 0.0958i)4-s + 0.447i·5-s + (−0.975 + 0.220i)6-s + 0.377i·7-s + (−0.989 − 0.143i)8-s + (0.857 − 0.515i)9-s + (0.0214 − 0.446i)10-s − 0.195·11-s + (0.984 − 0.173i)12-s + 1.37·13-s + (0.0181 − 0.377i)14-s + (0.119 + 0.430i)15-s + (0.981 + 0.190i)16-s + 1.66i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.984 - 0.173i$
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: \((2,\ 420,\ (\ :1/2),\ 0.984 - 0.173i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30725 + 0.114424i\)
\(L(\frac12)\) \(\approx\) \(1.30725 + 0.114424i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 + 0.0678i)T \)
3 \( 1 + (-1.66 + 0.463i)T \)
5 \( 1 - iT \)
7 \( 1 - iT \)
good11 \( 1 + 0.648T + 11T^{2} \)
13 \( 1 - 4.96T + 13T^{2} \)
17 \( 1 - 6.88iT - 17T^{2} \)
19 \( 1 + 4.22iT - 19T^{2} \)
23 \( 1 - 1.33T + 23T^{2} \)
29 \( 1 + 4.10iT - 29T^{2} \)
31 \( 1 - 7.63iT - 31T^{2} \)
37 \( 1 + 2.29T + 37T^{2} \)
41 \( 1 + 10.9iT - 41T^{2} \)
43 \( 1 - 5.13iT - 43T^{2} \)
47 \( 1 - 9.62T + 47T^{2} \)
53 \( 1 - 5.56iT - 53T^{2} \)
59 \( 1 - 0.421T + 59T^{2} \)
61 \( 1 - 0.179T + 61T^{2} \)
67 \( 1 + 10.4iT - 67T^{2} \)
71 \( 1 + 2.09T + 71T^{2} \)
73 \( 1 + 16.6T + 73T^{2} \)
79 \( 1 + 7.10iT - 79T^{2} \)
83 \( 1 + 0.766T + 83T^{2} \)
89 \( 1 + 11.5iT - 89T^{2} \)
97 \( 1 + 14.8T + 97T^{2} \)
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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

−10.83506612606533776386727131064, −10.37039287738550712409392741778, −9.053563117069560518006254356257, −8.665579775023225216419300908800, −7.76394356874230422093844398024, −6.78414189372229090069119919518, −5.94508724041056326952398103148, −3.84053174791414893565903887195, −2.78335884203708961430888791542, −1.54607741190527887295210502999, 1.28850135159307425256268759696, 2.79748064695315558965378229360, 3.99448151581065754582623769172, 5.51208131582099887057439860080, 6.87634341658694834028142801960, 7.76906858453580513931052863887, 8.517754629784795264983359957397, 9.278570613010494278720096266363, 10.01765389269259367831111630501, 10.91528135006498066030099947672

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