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Properties

Label 2-420-12.11-c1-0-4
Degree $2$
Conductor $420$
Sign $-0.526 - 0.850i$
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  + (−1.09 − 0.897i)2-s + (−1.62 + 0.606i)3-s + (0.389 + 1.96i)4-s + i·5-s + (2.31 + 0.792i)6-s + i·7-s + (1.33 − 2.49i)8-s + (2.26 − 1.96i)9-s + (0.897 − 1.09i)10-s + 1.07·11-s + (−1.82 − 2.94i)12-s − 1.72·13-s + (0.897 − 1.09i)14-s + (−0.606 − 1.62i)15-s + (−3.69 + 1.52i)16-s + 5.38i·17-s + ⋯
L(s)  = 1  + (−0.772 − 0.634i)2-s + (−0.936 + 0.350i)3-s + (0.194 + 0.980i)4-s + 0.447i·5-s + (0.946 + 0.323i)6-s + 0.377i·7-s + (0.471 − 0.881i)8-s + (0.754 − 0.656i)9-s + (0.283 − 0.345i)10-s + 0.323·11-s + (−0.526 − 0.850i)12-s − 0.478·13-s + (0.239 − 0.292i)14-s + (−0.156 − 0.418i)15-s + (−0.923 + 0.382i)16-s + 1.30i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.526 - 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.526 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.526 - 0.850i$
Analytic conductor: \(3.35371\)
Root analytic conductor: \(1.83131\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{420} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 420,\ (\ :1/2),\ -0.526 - 0.850i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.171389 + 0.307629i\)
\(L(\frac12)\) \(\approx\) \(0.171389 + 0.307629i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.09 + 0.897i)T \)
3 \( 1 + (1.62 - 0.606i)T \)
5 \( 1 - iT \)
7 \( 1 - iT \)
good11 \( 1 - 1.07T + 11T^{2} \)
13 \( 1 + 1.72T + 13T^{2} \)
17 \( 1 - 5.38iT - 17T^{2} \)
19 \( 1 + 2.00iT - 19T^{2} \)
23 \( 1 + 8.92T + 23T^{2} \)
29 \( 1 - 1.69iT - 29T^{2} \)
31 \( 1 - 3.27iT - 31T^{2} \)
37 \( 1 + 8.95T + 37T^{2} \)
41 \( 1 - 11.8iT - 41T^{2} \)
43 \( 1 + 2.46iT - 43T^{2} \)
47 \( 1 - 5.04T + 47T^{2} \)
53 \( 1 + 4.28iT - 53T^{2} \)
59 \( 1 + 12.3T + 59T^{2} \)
61 \( 1 + 11.1T + 61T^{2} \)
67 \( 1 - 4.42iT - 67T^{2} \)
71 \( 1 - 6.71T + 71T^{2} \)
73 \( 1 - 2.20T + 73T^{2} \)
79 \( 1 + 2.97iT - 79T^{2} \)
83 \( 1 + 9.69T + 83T^{2} \)
89 \( 1 - 8.62iT - 89T^{2} \)
97 \( 1 - 13.7T + 97T^{2} \)
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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

−11.38139147747524865576548246405, −10.51643608537003477059642713343, −9.988964521822151564383066425861, −9.010172186309662556930001226663, −7.948907202766751720974474615658, −6.82427987588965184155123699410, −5.97974435565838124899573776877, −4.50869374814684455836873291964, −3.41297127191896775147612326482, −1.78129404505019704616497483192, 0.32208335412294279824010510740, 1.87679910676057788921443418597, 4.35645270448696623533066958148, 5.38727024774499563845889872257, 6.22737609925493429023585871987, 7.26816012609002789570904414894, 7.85888977444396978983714270580, 9.124319804949698908170316855224, 9.978706671256063785919475726545, 10.71062708128974416173527930824

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