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Properties

Label 2-420-105.59-c1-0-5
Degree $2$
Conductor $420$
Sign $0.539 - 0.841i$
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  + (−1.16 + 1.28i)3-s + (1.81 + 1.30i)5-s + (2.08 − 1.63i)7-s + (−0.287 − 2.98i)9-s + (5.09 + 2.93i)11-s − 3.54·13-s + (−3.78 + 0.802i)15-s + (−2.69 − 1.55i)17-s + (3.58 − 2.06i)19-s + (−0.331 + 4.57i)21-s + (3.57 + 6.19i)23-s + (1.58 + 4.74i)25-s + (4.16 + 3.10i)27-s + 6.84i·29-s + (−3.90 − 2.25i)31-s + ⋯
L(s)  = 1  + (−0.672 + 0.740i)3-s + (0.811 + 0.584i)5-s + (0.786 − 0.617i)7-s + (−0.0959 − 0.995i)9-s + (1.53 + 0.886i)11-s − 0.981·13-s + (−0.978 + 0.207i)15-s + (−0.652 − 0.376i)17-s + (0.822 − 0.474i)19-s + (−0.0723 + 0.997i)21-s + (0.745 + 1.29i)23-s + (0.316 + 0.948i)25-s + (0.801 + 0.598i)27-s + 1.27i·29-s + (−0.701 − 0.404i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.539 - 0.841i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.539 - 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.539 - 0.841i$
Analytic conductor: \(3.35371\)
Root analytic conductor: \(1.83131\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{420} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 420,\ (\ :1/2),\ 0.539 - 0.841i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.23694 + 0.676177i\)
\(L(\frac12)\) \(\approx\) \(1.23694 + 0.676177i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.16 - 1.28i)T \)
5 \( 1 + (-1.81 - 1.30i)T \)
7 \( 1 + (-2.08 + 1.63i)T \)
good11 \( 1 + (-5.09 - 2.93i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 3.54T + 13T^{2} \)
17 \( 1 + (2.69 + 1.55i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.58 + 2.06i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.57 - 6.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 6.84iT - 29T^{2} \)
31 \( 1 + (3.90 + 2.25i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.70 - 0.986i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 6.33T + 41T^{2} \)
43 \( 1 - 3.88iT - 43T^{2} \)
47 \( 1 + (-1.58 + 0.916i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.90 + 11.9i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.32 + 4.02i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.702 + 0.405i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (8.84 + 5.10i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 1.18iT - 71T^{2} \)
73 \( 1 + (-1.08 + 1.87i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.05 + 8.75i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 5.31iT - 83T^{2} \)
89 \( 1 + (6.28 + 10.8i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 4.50T + 97T^{2} \)
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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

−11.38913009415351360142311516171, −10.40844453910474018319448864453, −9.594887183700598702155550538810, −9.083633340436085778864848779676, −7.15184946112669236879164147561, −6.86070367374660584773898137184, −5.38193288935222454423223402093, −4.68583017636921245238625360077, −3.44255287290468274461907745832, −1.61146045624605333839494530853, 1.18649834076851967841394859042, 2.35720743845468795554343950496, 4.45404276707807486504024191667, 5.47268887568044462886428351160, 6.17661597354132435956197091319, 7.18186571543020774623579945786, 8.477798479969997217704685277898, 9.018918088252492672081638191219, 10.24330372199463536218856665519, 11.28727943808696895452603322156

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