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Properties

Label 2-420-60.59-c1-0-41
Degree $2$
Conductor $420$
Sign $0.314 - 0.949i$
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  + (1.25 + 0.643i)2-s + (1.48 + 0.887i)3-s + (1.17 + 1.62i)4-s + (−2.05 − 0.887i)5-s + (1.30 + 2.07i)6-s + 7-s + (0.434 + 2.79i)8-s + (1.42 + 2.64i)9-s + (−2.01 − 2.43i)10-s + 2.75·11-s + (0.304 + 3.45i)12-s − 1.90i·13-s + (1.25 + 0.643i)14-s + (−2.26 − 3.14i)15-s + (−1.25 + 3.79i)16-s − 5.03·17-s + ⋯
L(s)  = 1  + (0.890 + 0.454i)2-s + (0.858 + 0.512i)3-s + (0.586 + 0.810i)4-s + (−0.917 − 0.397i)5-s + (0.531 + 0.847i)6-s + 0.377·7-s + (0.153 + 0.988i)8-s + (0.474 + 0.880i)9-s + (−0.636 − 0.771i)10-s + 0.832·11-s + (0.0880 + 0.996i)12-s − 0.529i·13-s + (0.336 + 0.171i)14-s + (−0.584 − 0.811i)15-s + (−0.312 + 0.949i)16-s − 1.22·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.314 - 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.314 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.314 - 0.949i$
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: \((2,\ 420,\ (\ :1/2),\ 0.314 - 0.949i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.20265 + 1.59017i\)
\(L(\frac12)\) \(\approx\) \(2.20265 + 1.59017i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.25 - 0.643i)T \)
3 \( 1 + (-1.48 - 0.887i)T \)
5 \( 1 + (2.05 + 0.887i)T \)
7 \( 1 - T \)
good11 \( 1 - 2.75T + 11T^{2} \)
13 \( 1 + 1.90iT - 13T^{2} \)
17 \( 1 + 5.03T + 17T^{2} \)
19 \( 1 - 1.90iT - 19T^{2} \)
23 \( 1 + 2.05iT - 23T^{2} \)
29 \( 1 + 4.94iT - 29T^{2} \)
31 \( 1 + 6.92iT - 31T^{2} \)
37 \( 1 + 6.48iT - 37T^{2} \)
41 \( 1 + 11.1iT - 41T^{2} \)
43 \( 1 + 9.36T + 43T^{2} \)
47 \( 1 - 7.51iT - 47T^{2} \)
53 \( 1 - 6.14T + 53T^{2} \)
59 \( 1 - 0.716T + 59T^{2} \)
61 \( 1 - 5.13T + 61T^{2} \)
67 \( 1 - 3.58T + 67T^{2} \)
71 \( 1 - 4.49T + 71T^{2} \)
73 \( 1 + 6.11iT - 73T^{2} \)
79 \( 1 - 14.6iT - 79T^{2} \)
83 \( 1 + 7.71iT - 83T^{2} \)
89 \( 1 - 13.5iT - 89T^{2} \)
97 \( 1 - 16.6iT - 97T^{2} \)
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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

−11.48260247472459473442962694509, −10.72241531170266136655064418298, −9.269641528430717596441726735828, −8.410041885008790241965728710005, −7.78825144143301132915331524990, −6.78513383200465141867206772322, −5.36439006920657564684345707332, −4.24551400741211081621254642998, −3.80546944218055177165752393624, −2.31768215658338337162775373949, 1.55981662479130125511244153315, 2.94047764889096207272452674132, 3.88957640610940389344515789261, 4.81314040650215671448272864062, 6.73446326926322150036034626573, 6.85821385103089731913447205583, 8.271811811735082047904514943721, 9.127122518783126245794102664722, 10.30722163986990431961791680681, 11.49596529688361952360623568894

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