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Properties

Label 2-420-420.83-c1-0-7
Degree $2$
Conductor $420$
Sign $-0.999 + 0.0260i$
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 + i)2-s + (1.22 + 1.22i)3-s − 2i·4-s + (−1.22 − 1.87i)5-s − 2.44·6-s + (−1.87 + 1.87i)7-s + (2 + 2i)8-s + 2.99i·9-s + (3.09 + 0.646i)10-s − 3.58·11-s + (2.44 − 2.44i)12-s − 3.74i·14-s + (0.791 − 3.79i)15-s − 4·16-s + (−5.54 + 5.54i)17-s + (−2.99 − 2.99i)18-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + (0.707 + 0.707i)3-s − i·4-s + (−0.547 − 0.836i)5-s − 0.999·6-s + (−0.707 + 0.707i)7-s + (0.707 + 0.707i)8-s + 0.999i·9-s + (0.978 + 0.204i)10-s − 1.08·11-s + (0.707 − 0.707i)12-s − 0.999i·14-s + (0.204 − 0.978i)15-s − 16-s + (−1.34 + 1.34i)17-s + (−0.707 − 0.707i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0260i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0260i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.999 + 0.0260i$
Analytic conductor: \(3.35371\)
Root analytic conductor: \(1.83131\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{420} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 420,\ (\ :1/2),\ -0.999 + 0.0260i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00695475 - 0.533609i\)
\(L(\frac12)\) \(\approx\) \(0.00695475 - 0.533609i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - i)T \)
3 \( 1 + (-1.22 - 1.22i)T \)
5 \( 1 + (1.22 + 1.87i)T \)
7 \( 1 + (1.87 - 1.87i)T \)
good11 \( 1 + 3.58T + 11T^{2} \)
13 \( 1 - 13iT^{2} \)
17 \( 1 + (5.54 - 5.54i)T - 17iT^{2} \)
19 \( 1 - 1.15iT - 19T^{2} \)
23 \( 1 + (-4.58 - 4.58i)T + 23iT^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 9.93T + 31T^{2} \)
37 \( 1 + (-8.58 + 8.58i)T - 37iT^{2} \)
41 \( 1 - 3.74T + 41T^{2} \)
43 \( 1 - 43iT^{2} \)
47 \( 1 + 47iT^{2} \)
53 \( 1 - 53iT^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 67iT^{2} \)
71 \( 1 - 6.41T + 71T^{2} \)
73 \( 1 - 73iT^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 83iT^{2} \)
89 \( 1 + 2.44iT - 89T^{2} \)
97 \( 1 + 97iT^{2} \)
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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

−11.23353207636320723733216057138, −10.57441275433354245271017673343, −9.356482289916967437592808656861, −9.029679169271077077027858669533, −8.153717935304558114310460859319, −7.38839363562173544864637339894, −5.87261401342179626859542133737, −5.05673679003158300997723780958, −3.84360415510670066324111612586, −2.18583346207856863540418543151, 0.37127806488961314713310430982, 2.51318630973689709965962594106, 3.10661032333012854242222008376, 4.37270891342791932872715023595, 6.66390252819599938138418415904, 7.18644078486464344237520478801, 7.933230412967241165686779023264, 8.986032627239052376229110668861, 9.805181394628411539194976695755, 10.84363622847354788067790097923

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