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Properties

Label 2-420-28.19-c1-0-15
Degree $2$
Conductor $420$
Sign $0.895 + 0.444i$
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 − i)2-s + (0.5 + 0.866i)3-s + 2i·4-s + (0.866 + 0.5i)5-s + (0.366 − 1.36i)6-s + (−0.5 − 2.59i)7-s + (2 − 2i)8-s + (−0.499 + 0.866i)9-s + (−0.366 − 1.36i)10-s + (1.73 − i)11-s + (−1.73 + i)12-s + 0.267i·13-s + (−2.09 + 3.09i)14-s + 0.999i·15-s − 4·16-s + (6 − 3.46i)17-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + (0.288 + 0.499i)3-s + i·4-s + (0.387 + 0.223i)5-s + (0.149 − 0.557i)6-s + (−0.188 − 0.981i)7-s + (0.707 − 0.707i)8-s + (−0.166 + 0.288i)9-s + (−0.115 − 0.431i)10-s + (0.522 − 0.301i)11-s + (−0.499 + 0.288i)12-s + 0.0743i·13-s + (−0.560 + 0.827i)14-s + 0.258i·15-s − 16-s + (1.45 − 0.840i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.895 + 0.444i$
Analytic conductor: \(3.35371\)
Root analytic conductor: \(1.83131\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{420} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 420,\ (\ :1/2),\ 0.895 + 0.444i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15090 - 0.269613i\)
\(L(\frac12)\) \(\approx\) \(1.15090 - 0.269613i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (0.5 + 2.59i)T \)
good11 \( 1 + (-1.73 + i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 0.267iT - 13T^{2} \)
17 \( 1 + (-6 + 3.46i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.23 - 2.13i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-6.92 - 4i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.53T + 29T^{2} \)
31 \( 1 + (-3.23 - 5.59i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.23 - 2.13i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 6iT - 41T^{2} \)
43 \( 1 + 9.19iT - 43T^{2} \)
47 \( 1 + (0.267 - 0.464i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.26 + 2.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.46 + 2.53i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (9.46 + 5.46i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.03 + 0.598i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 15.4iT - 71T^{2} \)
73 \( 1 + (-2.76 + 1.59i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.69 - 2.13i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 2T + 83T^{2} \)
89 \( 1 + (14.1 + 8.19i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 14.9iT - 97T^{2} \)
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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

−10.86150069585983515459440324690, −10.18029392276136892951750962885, −9.537721030322207815155020474988, −8.654002402801084944278032504672, −7.56567951674639029647512804182, −6.76530102180337152292914179157, −5.12531825747921367986760931213, −3.76394954278801020592040070017, −3.01810365918809937839150727925, −1.22525235011724995360000496144, 1.32767783393785903317366802082, 2.74907569056386376195936519070, 4.75911317266918689746839673193, 5.92073153062059669166155810561, 6.52407394181042980603428275582, 7.70316411877931333343373067411, 8.535981693873790921321025697593, 9.255835447024446140936758399733, 10.00609641079319354313957393711, 11.14783249992082228060930505620

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