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Properties

Label 2-420-140.139-c1-0-14
Degree $2$
Conductor $420$
Sign $0.512 - 0.858i$
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  + (−0.654 + 1.25i)2-s − i·3-s + (−1.14 − 1.64i)4-s + (0.206 + 2.22i)5-s + (1.25 + 0.654i)6-s + (−0.859 − 2.50i)7-s + (2.80 − 0.359i)8-s − 9-s + (−2.92 − 1.19i)10-s + 4.14i·11-s + (−1.64 + 1.14i)12-s + 3.35·13-s + (3.69 + 0.559i)14-s + (2.22 − 0.206i)15-s + (−1.38 + 3.75i)16-s + 3.18·17-s + ⋯
L(s)  = 1  + (−0.462 + 0.886i)2-s − 0.577i·3-s + (−0.571 − 0.820i)4-s + (0.0925 + 0.995i)5-s + (0.511 + 0.267i)6-s + (−0.324 − 0.945i)7-s + (0.991 − 0.127i)8-s − 0.333·9-s + (−0.925 − 0.378i)10-s + 1.24i·11-s + (−0.473 + 0.330i)12-s + 0.930·13-s + (0.988 + 0.149i)14-s + (0.574 − 0.0534i)15-s + (−0.346 + 0.938i)16-s + 0.773·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.512 - 0.858i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.512 - 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.512 - 0.858i$
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: \((2,\ 420,\ (\ :1/2),\ 0.512 - 0.858i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.928477 + 0.526845i\)
\(L(\frac12)\) \(\approx\) \(0.928477 + 0.526845i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.654 - 1.25i)T \)
3 \( 1 + iT \)
5 \( 1 + (-0.206 - 2.22i)T \)
7 \( 1 + (0.859 + 2.50i)T \)
good11 \( 1 - 4.14iT - 11T^{2} \)
13 \( 1 - 3.35T + 13T^{2} \)
17 \( 1 - 3.18T + 17T^{2} \)
19 \( 1 - 6.52T + 19T^{2} \)
23 \( 1 + 1.42T + 23T^{2} \)
29 \( 1 - 8.88T + 29T^{2} \)
31 \( 1 - 2.93T + 31T^{2} \)
37 \( 1 - 9.54iT - 37T^{2} \)
41 \( 1 - 3.55iT - 41T^{2} \)
43 \( 1 + 0.667T + 43T^{2} \)
47 \( 1 + 0.693iT - 47T^{2} \)
53 \( 1 + 4.18iT - 53T^{2} \)
59 \( 1 + 5.96T + 59T^{2} \)
61 \( 1 + 10.3iT - 61T^{2} \)
67 \( 1 + 7.05T + 67T^{2} \)
71 \( 1 - 9.50iT - 71T^{2} \)
73 \( 1 - 7.52T + 73T^{2} \)
79 \( 1 + 12.1iT - 79T^{2} \)
83 \( 1 + 5.25iT - 83T^{2} \)
89 \( 1 + 2.83iT - 89T^{2} \)
97 \( 1 + 15.9T + 97T^{2} \)
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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

−11.15362138931611958414078072347, −10.03096710534172224980716132158, −9.829926473785066843531865962785, −8.239430252310745060394632095159, −7.48724640274563811405653021099, −6.80805744154389268208243144831, −6.11069429630916693948681840731, −4.71095312477931933394115140259, −3.26107886021349568117686349671, −1.31662557027370762149685129042, 1.01818053400375206246509910632, 2.86526804694370584973260784458, 3.82455758053604339895208201441, 5.17235305877762955622185627101, 5.95984198475358984556259165587, 7.911789861048877218835245014012, 8.672410097422535340817947982049, 9.206156179043660519101103304351, 10.05719340719361584338036705612, 11.05899040973246727904918742077

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