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Properties

Label 2-420-60.59-c1-0-34
Degree $2$
Conductor $420$
Sign $0.673 + 0.739i$
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  + (−1.38 + 0.291i)2-s + (0.751 − 1.56i)3-s + (1.83 − 0.806i)4-s + (−1.60 + 1.56i)5-s + (−0.585 + 2.37i)6-s + 7-s + (−2.29 + 1.64i)8-s + (−1.86 − 2.34i)9-s + (1.76 − 2.62i)10-s + 4.26·11-s + (0.117 − 3.46i)12-s − 2.35i·13-s + (−1.38 + 0.291i)14-s + (1.23 + 3.67i)15-s + (2.70 − 2.95i)16-s + 5.53·17-s + ⋯
L(s)  = 1  + (−0.978 + 0.205i)2-s + (0.433 − 0.900i)3-s + (0.915 − 0.403i)4-s + (−0.716 + 0.697i)5-s + (−0.239 + 0.970i)6-s + 0.377·7-s + (−0.812 + 0.582i)8-s + (−0.623 − 0.781i)9-s + (0.557 − 0.830i)10-s + 1.28·11-s + (0.0340 − 0.999i)12-s − 0.654i·13-s + (−0.369 + 0.0778i)14-s + (0.317 + 0.948i)15-s + (0.675 − 0.737i)16-s + 1.34·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.673 + 0.739i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.673 + 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.673 + 0.739i$
Analytic conductor: \(3.35371\)
Root analytic conductor: \(1.83131\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{420} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 420,\ (\ :1/2),\ 0.673 + 0.739i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.916293 - 0.405049i\)
\(L(\frac12)\) \(\approx\) \(0.916293 - 0.405049i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.38 - 0.291i)T \)
3 \( 1 + (-0.751 + 1.56i)T \)
5 \( 1 + (1.60 - 1.56i)T \)
7 \( 1 - T \)
good11 \( 1 - 4.26T + 11T^{2} \)
13 \( 1 + 2.35iT - 13T^{2} \)
17 \( 1 - 5.53T + 17T^{2} \)
19 \( 1 - 2.35iT - 19T^{2} \)
23 \( 1 + 3.99iT - 23T^{2} \)
29 \( 1 + 7.60iT - 29T^{2} \)
31 \( 1 + 1.98iT - 31T^{2} \)
37 \( 1 - 3.22iT - 37T^{2} \)
41 \( 1 + 6.59iT - 41T^{2} \)
43 \( 1 - 8.36T + 43T^{2} \)
47 \( 1 - 8.77iT - 47T^{2} \)
53 \( 1 + 3.10T + 53T^{2} \)
59 \( 1 - 10.5T + 59T^{2} \)
61 \( 1 + 5.55T + 61T^{2} \)
67 \( 1 - 2.57T + 67T^{2} \)
71 \( 1 + 4.93T + 71T^{2} \)
73 \( 1 + 8.76iT - 73T^{2} \)
79 \( 1 + 2.79iT - 79T^{2} \)
83 \( 1 - 17.6iT - 83T^{2} \)
89 \( 1 + 1.70iT - 89T^{2} \)
97 \( 1 - 12.4iT - 97T^{2} \)
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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

−11.09023657422773076404313220815, −10.08546788185858462339644003022, −9.085817956843340424615881861713, −8.005652991793735973366233439236, −7.70660176896984099485008954251, −6.64268977628457033964445510147, −5.88688017384826875681802495150, −3.75657080033455057990204796291, −2.53725701894886734578096974770, −0.994568410609455101328297255979, 1.43394293806101866477975182461, 3.28469375197680713127087525895, 4.17293536861138248072185110641, 5.46301158331142358080676016073, 7.04043109337290080326985199129, 7.935246518561867018828671842925, 8.888756368152164207008191922138, 9.246298048408338917880988217121, 10.24536289626637417320635265438, 11.36555857678058668783311718469

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