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Properties

Label 2-420-140.139-c1-0-44
Degree $2$
Conductor $420$
Sign $-0.427 - 0.903i$
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  + (−1.08 − 0.903i)2-s − i·3-s + (0.366 + 1.96i)4-s + (−0.660 − 2.13i)5-s + (−0.903 + 1.08i)6-s + (−2.64 − 0.110i)7-s + (1.37 − 2.46i)8-s − 9-s + (−1.21 + 2.92i)10-s + 2.42i·11-s + (1.96 − 0.366i)12-s − 5.02·13-s + (2.77 + 2.50i)14-s + (−2.13 + 0.660i)15-s + (−3.73 + 1.43i)16-s + 6.95·17-s + ⋯
L(s)  = 1  + (−0.769 − 0.639i)2-s − 0.577i·3-s + (0.183 + 0.983i)4-s + (−0.295 − 0.955i)5-s + (−0.368 + 0.444i)6-s + (−0.999 − 0.0416i)7-s + (0.487 − 0.873i)8-s − 0.333·9-s + (−0.383 + 0.923i)10-s + 0.732i·11-s + (0.567 − 0.105i)12-s − 1.39·13-s + (0.741 + 0.670i)14-s + (−0.551 + 0.170i)15-s + (−0.932 + 0.359i)16-s + 1.68·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.427 - 0.903i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.427 - 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.427 - 0.903i$
Analytic conductor: \(3.35371\)
Root analytic conductor: \(1.83131\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{420} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 420,\ (\ :1/2),\ -0.427 - 0.903i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0589938 + 0.0931993i\)
\(L(\frac12)\) \(\approx\) \(0.0589938 + 0.0931993i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.08 + 0.903i)T \)
3 \( 1 + iT \)
5 \( 1 + (0.660 + 2.13i)T \)
7 \( 1 + (2.64 + 0.110i)T \)
good11 \( 1 - 2.42iT - 11T^{2} \)
13 \( 1 + 5.02T + 13T^{2} \)
17 \( 1 - 6.95T + 17T^{2} \)
19 \( 1 + 1.26T + 19T^{2} \)
23 \( 1 + 6.81T + 23T^{2} \)
29 \( 1 + 4.56T + 29T^{2} \)
31 \( 1 + 0.903T + 31T^{2} \)
37 \( 1 - 8.02iT - 37T^{2} \)
41 \( 1 + 3.33iT - 41T^{2} \)
43 \( 1 - 2.06T + 43T^{2} \)
47 \( 1 + 3.31iT - 47T^{2} \)
53 \( 1 - 4.77iT - 53T^{2} \)
59 \( 1 + 8.38T + 59T^{2} \)
61 \( 1 - 4.38iT - 61T^{2} \)
67 \( 1 + 8.25T + 67T^{2} \)
71 \( 1 + 11.5iT - 71T^{2} \)
73 \( 1 + 12.3T + 73T^{2} \)
79 \( 1 + 8.69iT - 79T^{2} \)
83 \( 1 + 9.72iT - 83T^{2} \)
89 \( 1 + 13.6iT - 89T^{2} \)
97 \( 1 + 1.23T + 97T^{2} \)
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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

−10.24019768931272509162026174004, −9.765526804831298311697106468250, −8.903297316884944378088530727828, −7.72681740155204156391479793402, −7.34258320971784153178641074280, −5.89830136120852994237256342976, −4.45459908767820884953838686440, −3.17125634787948160350154189883, −1.76292388618026847248258046458, −0.086115767735784519804755726820, 2.61908403401620577263816504550, 3.83068166687450105817972233195, 5.49541605765500086321962328122, 6.21449442115669925604448413979, 7.33397757268742514198252908901, 7.984334801684631087758265063447, 9.323710829548692952939584004532, 9.937916789502864817397390670305, 10.51755919357561172995935940859, 11.53111700887271129016431835910

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