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Properties

Label 2-420-28.27-c1-0-23
Degree $2$
Conductor $420$
Sign $-0.996 + 0.0891i$
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  + (−0.449 − 1.34i)2-s − 3-s + (−1.59 + 1.20i)4-s − i·5-s + (0.449 + 1.34i)6-s + (1.40 − 2.24i)7-s + (2.33 + 1.59i)8-s + 9-s + (−1.34 + 0.449i)10-s − 3.99i·11-s + (1.59 − 1.20i)12-s + 2.50i·13-s + (−3.63 − 0.868i)14-s + i·15-s + (1.09 − 3.84i)16-s + 1.07i·17-s + ⋯
L(s)  = 1  + (−0.317 − 0.948i)2-s − 0.577·3-s + (−0.797 + 0.602i)4-s − 0.447i·5-s + (0.183 + 0.547i)6-s + (0.529 − 0.848i)7-s + (0.825 + 0.564i)8-s + 0.333·9-s + (−0.424 + 0.142i)10-s − 1.20i·11-s + (0.460 − 0.348i)12-s + 0.694i·13-s + (−0.972 − 0.232i)14-s + 0.258i·15-s + (0.273 − 0.961i)16-s + 0.261i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0891i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 + 0.0891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.996 + 0.0891i$
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: \((2,\ 420,\ (\ :1/2),\ -0.996 + 0.0891i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0296796 - 0.664170i\)
\(L(\frac12)\) \(\approx\) \(0.0296796 - 0.664170i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.449 + 1.34i)T \)
3 \( 1 + T \)
5 \( 1 + iT \)
7 \( 1 + (-1.40 + 2.24i)T \)
good11 \( 1 + 3.99iT - 11T^{2} \)
13 \( 1 - 2.50iT - 13T^{2} \)
17 \( 1 - 1.07iT - 17T^{2} \)
19 \( 1 + 7.78T + 19T^{2} \)
23 \( 1 + 8.10iT - 23T^{2} \)
29 \( 1 + 5.99T + 29T^{2} \)
31 \( 1 + 6.32T + 31T^{2} \)
37 \( 1 - 3.05T + 37T^{2} \)
41 \( 1 + 9.32iT - 41T^{2} \)
43 \( 1 - 8.12iT - 43T^{2} \)
47 \( 1 - 9.01T + 47T^{2} \)
53 \( 1 - 2.86T + 53T^{2} \)
59 \( 1 - 9.68T + 59T^{2} \)
61 \( 1 + 3.21iT - 61T^{2} \)
67 \( 1 + 7.79iT - 67T^{2} \)
71 \( 1 - 5.84iT - 71T^{2} \)
73 \( 1 - 5.73iT - 73T^{2} \)
79 \( 1 + 2.81iT - 79T^{2} \)
83 \( 1 + 12.5T + 83T^{2} \)
89 \( 1 + 3.51iT - 89T^{2} \)
97 \( 1 + 12.0iT - 97T^{2} \)
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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

−10.97732392170016882861796976331, −10.16947284609596954677261256667, −8.891685340981995546738494955675, −8.357892836467613096605503743336, −7.13708754362167019128529549396, −5.84302155762425309208446671841, −4.52422614041194493805668934976, −3.88250975621343125816586091916, −2.02032874379788518200133775055, −0.51220697244267733434469105704, 1.95724435614507525813476274525, 4.09660174090813399872469843630, 5.25503920329395191627839737642, 5.91098830844441048619176641390, 7.06516164163538914579931354797, 7.73856038298783421874555075431, 8.863983306275984908191870480663, 9.732563387906245105680485754273, 10.61945472281962897105518949300, 11.49801395265138925850870008126

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