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Properties

Label 2-420-105.2-c1-0-4
Degree $2$
Conductor $420$
Sign $0.945 + 0.324i$
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  + (−1.56 − 0.735i)3-s + (−1.07 + 1.96i)5-s + (−2.28 − 1.34i)7-s + (1.91 + 2.30i)9-s + (4.85 − 2.80i)11-s + (0.354 − 0.354i)13-s + (3.12 − 2.28i)15-s + (3.57 + 0.959i)17-s + (2.40 + 1.39i)19-s + (2.59 + 3.78i)21-s + (3.95 − 1.06i)23-s + (−2.68 − 4.21i)25-s + (−1.30 − 5.02i)27-s + 8.44·29-s + (−0.730 − 1.26i)31-s + ⋯
L(s)  = 1  + (−0.905 − 0.424i)3-s + (−0.480 + 0.876i)5-s + (−0.862 − 0.506i)7-s + (0.639 + 0.769i)9-s + (1.46 − 0.845i)11-s + (0.0984 − 0.0984i)13-s + (0.807 − 0.589i)15-s + (0.868 + 0.232i)17-s + (0.552 + 0.319i)19-s + (0.565 + 0.824i)21-s + (0.825 − 0.221i)23-s + (−0.537 − 0.842i)25-s + (−0.252 − 0.967i)27-s + 1.56·29-s + (−0.131 − 0.227i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 + 0.324i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.945 + 0.324i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.945 + 0.324i$
Analytic conductor: \(3.35371\)
Root analytic conductor: \(1.83131\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{420} (317, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 420,\ (\ :1/2),\ 0.945 + 0.324i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.936076 - 0.156300i\)
\(L(\frac12)\) \(\approx\) \(0.936076 - 0.156300i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.56 + 0.735i)T \)
5 \( 1 + (1.07 - 1.96i)T \)
7 \( 1 + (2.28 + 1.34i)T \)
good11 \( 1 + (-4.85 + 2.80i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.354 + 0.354i)T - 13iT^{2} \)
17 \( 1 + (-3.57 - 0.959i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-2.40 - 1.39i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.95 + 1.06i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 8.44T + 29T^{2} \)
31 \( 1 + (0.730 + 1.26i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (7.29 - 1.95i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 1.89iT - 41T^{2} \)
43 \( 1 + (-7.19 + 7.19i)T - 43iT^{2} \)
47 \( 1 + (-2.88 - 10.7i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.636 + 2.37i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (5.64 + 9.78i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.67 + 2.90i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.921 - 3.43i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 10.5iT - 71T^{2} \)
73 \( 1 + (-10.4 - 2.79i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (1.42 + 0.822i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.481 + 0.481i)T + 83iT^{2} \)
89 \( 1 + (7.37 - 12.7i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.38 + 5.38i)T + 97iT^{2} \)
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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

−11.12362514305104014759335898727, −10.49310704995812881053838896817, −9.560147927106316847765218939742, −8.215913354893879656358618210521, −7.08122862286707833656328696547, −6.57630004133504985199253594975, −5.70065560676445249975869273351, −4.09463153792415284401770734922, −3.14544857167346777852211064486, −0.955722195606856734182107360453, 1.09766897396592814425617436385, 3.43542834680725304184682807478, 4.48645477042714043364681141169, 5.39768728160486373599310522622, 6.47975554764820987171115214763, 7.31488366877786148870877314743, 8.904962299667388815172733071080, 9.376212989570565353715223038116, 10.24364046175475029603568946424, 11.50614582399523693214022145909

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