Properties

Label 2-420-28.27-c1-0-22
Degree $2$
Conductor $420$
Sign $0.894 + 0.447i$
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  + (1.40 + 0.117i)2-s − 3-s + (1.97 + 0.329i)4-s i·5-s + (−1.40 − 0.117i)6-s + (0.776 − 2.52i)7-s + (2.74 + 0.695i)8-s + 9-s + (0.117 − 1.40i)10-s − 0.556i·11-s + (−1.97 − 0.329i)12-s + 0.182i·13-s + (1.38 − 3.47i)14-s + i·15-s + (3.78 + 1.30i)16-s − 2.39i·17-s + ⋯
L(s)  = 1  + (0.996 + 0.0827i)2-s − 0.577·3-s + (0.986 + 0.164i)4-s − 0.447i·5-s + (−0.575 − 0.0477i)6-s + (0.293 − 0.956i)7-s + (0.969 + 0.246i)8-s + 0.333·9-s + (0.0370 − 0.445i)10-s − 0.167i·11-s + (−0.569 − 0.0952i)12-s + 0.0505i·13-s + (0.371 − 0.928i)14-s + 0.258i·15-s + (0.945 + 0.325i)16-s − 0.580i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.894 + 0.447i$
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: \((2,\ 420,\ (\ :1/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.17954 - 0.514318i\)
\(L(\frac12)\) \(\approx\) \(2.17954 - 0.514318i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 - 0.117i)T \)
3 \( 1 + T \)
5 \( 1 + iT \)
7 \( 1 + (-0.776 + 2.52i)T \)
good11 \( 1 + 0.556iT - 11T^{2} \)
13 \( 1 - 0.182iT - 13T^{2} \)
17 \( 1 + 2.39iT - 17T^{2} \)
19 \( 1 - 3.08T + 19T^{2} \)
23 \( 1 - 3.94iT - 23T^{2} \)
29 \( 1 + 3.20T + 29T^{2} \)
31 \( 1 - 5.68T + 31T^{2} \)
37 \( 1 + 4.98T + 37T^{2} \)
41 \( 1 - 9.64iT - 41T^{2} \)
43 \( 1 - 0.643iT - 43T^{2} \)
47 \( 1 + 3.63T + 47T^{2} \)
53 \( 1 + 6.97T + 53T^{2} \)
59 \( 1 + 8.79T + 59T^{2} \)
61 \( 1 - 14.3iT - 61T^{2} \)
67 \( 1 + 10.0iT - 67T^{2} \)
71 \( 1 - 1.36iT - 71T^{2} \)
73 \( 1 + 10.1iT - 73T^{2} \)
79 \( 1 - 13.0iT - 79T^{2} \)
83 \( 1 + 9.45T + 83T^{2} \)
89 \( 1 + 8.01iT - 89T^{2} \)
97 \( 1 + 0.445iT - 97T^{2} \)
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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

−11.41391082172705880039705689708, −10.50841126636632361080528119567, −9.550046934286846171798270400003, −7.992546147709042410096122818639, −7.25580133025427542805106661891, −6.25034828110398219054342082773, −5.17766083469080327742055039999, −4.45557421211286439855967002100, −3.27128715977452165964570632130, −1.36794876585859532499764946031, 1.93113602641754270590606672714, 3.22085010000612524207595116672, 4.55950608704857254266284925170, 5.51270862378189592117991957165, 6.28105984969235203903554149748, 7.23446181908725064705396784572, 8.366852541929243825684057902514, 9.754250126706708278001590876914, 10.69006431599885842863472524357, 11.41390542190745902301180395096

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