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Properties

Label 2-420-28.27-c1-0-15
Degree $2$
Conductor $420$
Sign $0.812 - 0.582i$
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  + (−0.947 + 1.04i)2-s + 3-s + (−0.204 − 1.98i)4-s + i·5-s + (−0.947 + 1.04i)6-s + (2.29 − 1.31i)7-s + (2.28 + 1.66i)8-s + 9-s + (−1.04 − 0.947i)10-s + 0.477i·11-s + (−0.204 − 1.98i)12-s − 2.96i·13-s + (−0.796 + 3.65i)14-s + i·15-s + (−3.91 + 0.814i)16-s − 3.83i·17-s + ⋯
L(s)  = 1  + (−0.669 + 0.742i)2-s + 0.577·3-s + (−0.102 − 0.994i)4-s + 0.447i·5-s + (−0.386 + 0.428i)6-s + (0.868 − 0.496i)7-s + (0.807 + 0.590i)8-s + 0.333·9-s + (−0.332 − 0.299i)10-s + 0.143i·11-s + (−0.0590 − 0.574i)12-s − 0.821i·13-s + (−0.212 + 0.977i)14-s + 0.258i·15-s + (−0.979 + 0.203i)16-s − 0.929i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.812 - 0.582i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.812 - 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.812 - 0.582i$
Analytic conductor: \(3.35371\)
Root analytic conductor: \(1.83131\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{420} (391, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 420,\ (\ :1/2),\ 0.812 - 0.582i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26687 + 0.407233i\)
\(L(\frac12)\) \(\approx\) \(1.26687 + 0.407233i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.947 - 1.04i)T \)
3 \( 1 - T \)
5 \( 1 - iT \)
7 \( 1 + (-2.29 + 1.31i)T \)
good11 \( 1 - 0.477iT - 11T^{2} \)
13 \( 1 + 2.96iT - 13T^{2} \)
17 \( 1 + 3.83iT - 17T^{2} \)
19 \( 1 - 5.31T + 19T^{2} \)
23 \( 1 - 7.60iT - 23T^{2} \)
29 \( 1 - 6.17T + 29T^{2} \)
31 \( 1 - 3.38T + 31T^{2} \)
37 \( 1 + 8.62T + 37T^{2} \)
41 \( 1 - 1.01iT - 41T^{2} \)
43 \( 1 - 6.85iT - 43T^{2} \)
47 \( 1 - 6.21T + 47T^{2} \)
53 \( 1 + 9.30T + 53T^{2} \)
59 \( 1 + 4.88T + 59T^{2} \)
61 \( 1 - 4.75iT - 61T^{2} \)
67 \( 1 + 1.30iT - 67T^{2} \)
71 \( 1 - 9.18iT - 71T^{2} \)
73 \( 1 + 4.49iT - 73T^{2} \)
79 \( 1 + 8.80iT - 79T^{2} \)
83 \( 1 + 10.9T + 83T^{2} \)
89 \( 1 + 13.1iT - 89T^{2} \)
97 \( 1 + 1.60iT - 97T^{2} \)
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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

−11.04314918003645835844363492807, −10.13607800072777373924014664160, −9.463684965476655453938935326428, −8.341999860184492736006023345635, −7.59518229765029395292285988767, −7.04536620197697021548070907217, −5.61959145041877763526449624843, −4.67219347025584588436996675605, −3.04831083084452545139409055739, −1.33251450773711396630767582563, 1.42632073323977938329004352148, 2.59032065535142308579168852566, 3.98882067592135636341816741750, 4.99210242366314358064954715604, 6.66354656770403738989995921717, 7.889943290214056914012058724198, 8.544909136318783647485149056160, 9.129490608103991115198999127970, 10.19369617064886711864986911804, 11.01490986977019013469864236872

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