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Properties

Label 2-420-12.11-c1-0-0
Degree $2$
Conductor $420$
Sign $-0.655 - 0.754i$
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  + (−1.38 + 0.299i)2-s + (−0.491 − 1.66i)3-s + (1.82 − 0.829i)4-s + i·5-s + (1.17 + 2.14i)6-s + i·7-s + (−2.26 + 1.69i)8-s + (−2.51 + 1.63i)9-s + (−0.299 − 1.38i)10-s − 1.79·11-s + (−2.27 − 2.61i)12-s − 4.13·13-s + (−0.299 − 1.38i)14-s + (1.66 − 0.491i)15-s + (2.62 − 3.01i)16-s − 3.20i·17-s + ⋯
L(s)  = 1  + (−0.977 + 0.212i)2-s + (−0.283 − 0.958i)3-s + (0.910 − 0.414i)4-s + 0.447i·5-s + (0.480 + 0.876i)6-s + 0.377i·7-s + (−0.801 + 0.598i)8-s + (−0.838 + 0.544i)9-s + (−0.0948 − 0.437i)10-s − 0.540·11-s + (−0.655 − 0.754i)12-s − 1.14·13-s + (−0.0801 − 0.369i)14-s + (0.428 − 0.126i)15-s + (0.656 − 0.754i)16-s − 0.777i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.655 - 0.754i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.655 - 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.655 - 0.754i$
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: \((2,\ 420,\ (\ :1/2),\ -0.655 - 0.754i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0867734 + 0.190332i\)
\(L(\frac12)\) \(\approx\) \(0.0867734 + 0.190332i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.38 - 0.299i)T \)
3 \( 1 + (0.491 + 1.66i)T \)
5 \( 1 - iT \)
7 \( 1 - iT \)
good11 \( 1 + 1.79T + 11T^{2} \)
13 \( 1 + 4.13T + 13T^{2} \)
17 \( 1 + 3.20iT - 17T^{2} \)
19 \( 1 - 5.62iT - 19T^{2} \)
23 \( 1 + 6.63T + 23T^{2} \)
29 \( 1 - 5.01iT - 29T^{2} \)
31 \( 1 + 0.158iT - 31T^{2} \)
37 \( 1 + 5.00T + 37T^{2} \)
41 \( 1 - 2.64iT - 41T^{2} \)
43 \( 1 - 5.69iT - 43T^{2} \)
47 \( 1 + 4.36T + 47T^{2} \)
53 \( 1 + 4.06iT - 53T^{2} \)
59 \( 1 - 5.93T + 59T^{2} \)
61 \( 1 - 8.35T + 61T^{2} \)
67 \( 1 - 11.9iT - 67T^{2} \)
71 \( 1 + 14.4T + 71T^{2} \)
73 \( 1 + 13.5T + 73T^{2} \)
79 \( 1 + 7.04iT - 79T^{2} \)
83 \( 1 - 12.5T + 83T^{2} \)
89 \( 1 + 4.18iT - 89T^{2} \)
97 \( 1 - 15.3T + 97T^{2} \)
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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

−11.65775212761231103418396984019, −10.45045172778409420791923122479, −9.828358303893147384508817188573, −8.571794813559632016136583994016, −7.75394636186406529196209639170, −7.11091134846436735500937572166, −6.11010525394433000236034894589, −5.24371865954760312786209437432, −2.89602046532949266015242599181, −1.85112515717540700600992683208, 0.17429376962045651424259541653, 2.37499188629783093479154468240, 3.81607692305528745113202976462, 4.97801335967525842691536197077, 6.16070019203938932628342710341, 7.37988093143058417906097975090, 8.346745368951850632988714834681, 9.205050211428724314327624074880, 10.07184967039105225989416566200, 10.52396492342454365169211679991

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