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Properties

Label 2-420-105.2-c1-0-8
Degree $2$
Conductor $420$
Sign $0.906 + 0.422i$
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  + (1.44 − 0.954i)3-s + (−2.17 + 0.530i)5-s + (1.78 + 1.95i)7-s + (1.17 − 2.75i)9-s + (2.07 − 1.19i)11-s + (3.37 − 3.37i)13-s + (−2.63 + 2.84i)15-s + (4.74 + 1.27i)17-s + (0.151 + 0.0873i)19-s + (4.44 + 1.11i)21-s + (−1.21 + 0.324i)23-s + (4.43 − 2.30i)25-s + (−0.927 − 5.11i)27-s − 4.65·29-s + (1.26 + 2.18i)31-s + ⋯
L(s)  = 1  + (0.834 − 0.550i)3-s + (−0.971 + 0.237i)5-s + (0.675 + 0.737i)7-s + (0.393 − 0.919i)9-s + (0.626 − 0.361i)11-s + (0.937 − 0.937i)13-s + (−0.679 + 0.733i)15-s + (1.15 + 0.308i)17-s + (0.0346 + 0.0200i)19-s + (0.970 + 0.243i)21-s + (−0.252 + 0.0676i)23-s + (0.887 − 0.461i)25-s + (−0.178 − 0.983i)27-s − 0.863·29-s + (0.226 + 0.392i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.906 + 0.422i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.906 + 0.422i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.906 + 0.422i$
Analytic conductor: \(3.35371\)
Root analytic conductor: \(1.83131\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{420} (317, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 420,\ (\ :1/2),\ 0.906 + 0.422i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.72588 - 0.382307i\)
\(L(\frac12)\) \(\approx\) \(1.72588 - 0.382307i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.44 + 0.954i)T \)
5 \( 1 + (2.17 - 0.530i)T \)
7 \( 1 + (-1.78 - 1.95i)T \)
good11 \( 1 + (-2.07 + 1.19i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.37 + 3.37i)T - 13iT^{2} \)
17 \( 1 + (-4.74 - 1.27i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-0.151 - 0.0873i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.21 - 0.324i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 4.65T + 29T^{2} \)
31 \( 1 + (-1.26 - 2.18i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (10.9 - 2.94i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 1.78iT - 41T^{2} \)
43 \( 1 + (2.46 - 2.46i)T - 43iT^{2} \)
47 \( 1 + (2.07 + 7.73i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (2.92 - 10.8i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-2.06 - 3.57i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.47 + 6.02i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.12 + 11.6i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 15.9iT - 71T^{2} \)
73 \( 1 + (5.23 + 1.40i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-5.52 - 3.18i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.05 + 3.05i)T + 83iT^{2} \)
89 \( 1 + (2.58 - 4.47i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.70 + 2.70i)T + 97iT^{2} \)
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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

−11.33175461292367427350627517046, −10.26025108555590658655434341991, −8.948784121862764762005167631106, −8.288215827408865658870071134636, −7.73617011731069759095413318224, −6.58324394122330001722347062684, −5.43836640567075201930574632207, −3.83259330209452048981094269541, −3.08406722900654530049893700769, −1.38828428991876060832590978471, 1.59160331774032693177510834298, 3.56442465288087186932684885668, 4.09095936401919038293676078498, 5.13781639680122909784035349612, 6.91904636637226707716625057647, 7.71971147709983409887259051281, 8.503221185690444429722700966716, 9.324286370352205964869902144049, 10.35294070550029459681259887872, 11.25939579784602815635598275090

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