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Properties

Label 2-420-28.27-c1-0-28
Degree $2$
Conductor $420$
Sign $-0.399 + 0.916i$
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  + (0.309 − 1.37i)2-s + 3-s + (−1.80 − 0.853i)4-s − i·5-s + (0.309 − 1.37i)6-s + (2.64 − 0.0785i)7-s + (−1.73 + 2.23i)8-s + 9-s + (−1.37 − 0.309i)10-s − 0.987i·11-s + (−1.80 − 0.853i)12-s − 4.69i·13-s + (0.709 − 3.67i)14-s − i·15-s + (2.54 + 3.08i)16-s − 3.93i·17-s + ⋯
L(s)  = 1  + (0.218 − 0.975i)2-s + 0.577·3-s + (−0.904 − 0.426i)4-s − 0.447i·5-s + (0.126 − 0.563i)6-s + (0.999 − 0.0296i)7-s + (−0.614 + 0.789i)8-s + 0.333·9-s + (−0.436 − 0.0977i)10-s − 0.297i·11-s + (−0.522 − 0.246i)12-s − 1.30i·13-s + (0.189 − 0.981i)14-s − 0.258i·15-s + (0.635 + 0.771i)16-s − 0.954i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.399 + 0.916i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.399 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.399 + 0.916i$
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: \((2,\ 420,\ (\ :1/2),\ -0.399 + 0.916i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.979356 - 1.49540i\)
\(L(\frac12)\) \(\approx\) \(0.979356 - 1.49540i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 + 1.37i)T \)
3 \( 1 - T \)
5 \( 1 + iT \)
7 \( 1 + (-2.64 + 0.0785i)T \)
good11 \( 1 + 0.987iT - 11T^{2} \)
13 \( 1 + 4.69iT - 13T^{2} \)
17 \( 1 + 3.93iT - 17T^{2} \)
19 \( 1 + 0.223T + 19T^{2} \)
23 \( 1 - 5.88iT - 23T^{2} \)
29 \( 1 + 10.2T + 29T^{2} \)
31 \( 1 - 2.77T + 31T^{2} \)
37 \( 1 - 8.26T + 37T^{2} \)
41 \( 1 - 7.34iT - 41T^{2} \)
43 \( 1 - 4.32iT - 43T^{2} \)
47 \( 1 - 2.40T + 47T^{2} \)
53 \( 1 + 8.35T + 53T^{2} \)
59 \( 1 - 13.9T + 59T^{2} \)
61 \( 1 + 4.93iT - 61T^{2} \)
67 \( 1 - 7.84iT - 67T^{2} \)
71 \( 1 + 8.49iT - 71T^{2} \)
73 \( 1 - 14.4iT - 73T^{2} \)
79 \( 1 - 11.5iT - 79T^{2} \)
83 \( 1 + 1.67T + 83T^{2} \)
89 \( 1 + 0.493iT - 89T^{2} \)
97 \( 1 - 4.31iT - 97T^{2} \)
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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

−11.18637053154307323589054127686, −9.952479639406503996082166850868, −9.289415189629131913287578401628, −8.244026110896239550879765631940, −7.64479543366023548422023508187, −5.68765864821129632167421378818, −4.91905490190297155200917975973, −3.75651303811333661331841845726, −2.58519639144915520094548214525, −1.16360975656584479367777353260, 2.09124470444325012590511219387, 3.85227022824274035175973062792, 4.59853035664558202451404338840, 5.91537550301532491687212267206, 6.93853124879651555751218559838, 7.72721804731522825725432860855, 8.594316186029348601624697621756, 9.331296515634044965056690969754, 10.47468135026714814033845294069, 11.54349795722472866623569215925

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