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Properties

Label 2-420-21.17-c1-0-6
Degree $2$
Conductor $420$
Sign $0.368 + 0.929i$
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  + (−0.747 + 1.56i)3-s + (−0.5 − 0.866i)5-s + (−0.456 − 2.60i)7-s + (−1.88 − 2.33i)9-s + (−0.698 − 0.403i)11-s − 3.86i·13-s + (1.72 − 0.133i)15-s + (1.05 − 1.83i)17-s + (2.70 − 1.56i)19-s + (4.41 + 1.23i)21-s + (−4.21 + 2.43i)23-s + (−0.499 + 0.866i)25-s + (5.05 − 1.19i)27-s − 6.67i·29-s + (5.65 + 3.26i)31-s + ⋯
L(s)  = 1  + (−0.431 + 0.902i)3-s + (−0.223 − 0.387i)5-s + (−0.172 − 0.985i)7-s + (−0.627 − 0.778i)9-s + (−0.210 − 0.121i)11-s − 1.07i·13-s + (0.445 − 0.0344i)15-s + (0.257 − 0.445i)17-s + (0.620 − 0.358i)19-s + (0.962 + 0.269i)21-s + (−0.879 + 0.508i)23-s + (−0.0999 + 0.173i)25-s + (0.973 − 0.229i)27-s − 1.23i·29-s + (1.01 + 0.585i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.368 + 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.368 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.368 + 0.929i$
Analytic conductor: \(3.35371\)
Root analytic conductor: \(1.83131\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{420} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 420,\ (\ :1/2),\ 0.368 + 0.929i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.717567 - 0.487467i\)
\(L(\frac12)\) \(\approx\) \(0.717567 - 0.487467i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.747 - 1.56i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (0.456 + 2.60i)T \)
good11 \( 1 + (0.698 + 0.403i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 3.86iT - 13T^{2} \)
17 \( 1 + (-1.05 + 1.83i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.70 + 1.56i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.21 - 2.43i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 6.67iT - 29T^{2} \)
31 \( 1 + (-5.65 - 3.26i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.89 + 6.75i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 8.44T + 41T^{2} \)
43 \( 1 + 0.819T + 43T^{2} \)
47 \( 1 + (1.38 + 2.40i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (11.5 + 6.67i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.86 - 4.95i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.79 + 1.03i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.45 - 9.44i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 10.3iT - 71T^{2} \)
73 \( 1 + (-2.67 - 1.54i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.76 - 11.7i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 12.8T + 83T^{2} \)
89 \( 1 + (-1.60 - 2.78i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 1.01iT - 97T^{2} \)
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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

−10.93741975303376714265305076842, −10.10053033256036988274017324338, −9.518007079472830523773637221826, −8.264949882217376984956087283410, −7.41005336049430969865568165101, −6.07461126220582953894830687120, −5.13140110802744656348164793906, −4.14257039098701269123086957760, −3.13520945709406000985735004845, −0.59359032789607500663726377629, 1.78417826824832018696396861679, 3.02161382165600285796082555804, 4.70431235086130777012551701606, 5.93293744916433850522306133446, 6.55484119849921925071366419799, 7.64865048676129087624437115089, 8.457874943124310336886397616982, 9.535438373408198840370397060337, 10.64824211637295656969191710635, 11.61395447523534478666250983578

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