Properties

Label 2-420-60.59-c1-0-61
Degree $2$
Conductor $420$
Sign $-0.289 + 0.957i$
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  + 1.41i·2-s + (1 − 1.41i)3-s − 2.00·4-s + (−2.12 + 0.707i)5-s + (2.00 + 1.41i)6-s − 7-s − 2.82i·8-s + (−1.00 − 2.82i)9-s + (−1.00 − 3i)10-s − 4.24·11-s + (−2.00 + 2.82i)12-s − 6i·13-s − 1.41i·14-s + (−1.12 + 3.70i)15-s + 4.00·16-s − 4.24·17-s + ⋯
L(s)  = 1  + 0.999i·2-s + (0.577 − 0.816i)3-s − 1.00·4-s + (−0.948 + 0.316i)5-s + (0.816 + 0.577i)6-s − 0.377·7-s − 1.00i·8-s + (−0.333 − 0.942i)9-s + (−0.316 − 0.948i)10-s − 1.27·11-s + (−0.577 + 0.816i)12-s − 1.66i·13-s − 0.377i·14-s + (−0.289 + 0.957i)15-s + 1.00·16-s − 1.02·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.289 + 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.289 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.289 + 0.957i$
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: \((2,\ 420,\ (\ :1/2),\ -0.289 + 0.957i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.264959 - 0.356959i\)
\(L(\frac12)\) \(\approx\) \(0.264959 - 0.356959i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41iT \)
3 \( 1 + (-1 + 1.41i)T \)
5 \( 1 + (2.12 - 0.707i)T \)
7 \( 1 + T \)
good11 \( 1 + 4.24T + 11T^{2} \)
13 \( 1 + 6iT - 13T^{2} \)
17 \( 1 + 4.24T + 17T^{2} \)
19 \( 1 - 6iT - 19T^{2} \)
23 \( 1 + 1.41iT - 23T^{2} \)
29 \( 1 + 2.82iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + 6iT - 37T^{2} \)
41 \( 1 - 1.41iT - 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 - 2.82iT - 47T^{2} \)
53 \( 1 - 8.48T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 - 12.7T + 71T^{2} \)
73 \( 1 - 6iT - 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 - 2.82iT - 83T^{2} \)
89 \( 1 + 7.07iT - 89T^{2} \)
97 \( 1 - 6iT - 97T^{2} \)
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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

−10.77933323805184517415785779894, −9.901506924633228072121532306155, −8.541539581724183848424596182802, −8.002561175064726616678931374075, −7.43033369882296846625553897722, −6.39966139487070147754717295273, −5.40201733051104344411838326159, −3.89468136903113758750165607966, −2.84630250078274688643267299212, −0.25553502174852322392923186005, 2.31083557992343887396135033025, 3.41152357289031347451389212760, 4.47149193766339600647284700926, 5.02907663968505821647710057388, 7.01973922301509689873911774398, 8.260875174384469127443053910521, 8.923413116665325856527383317172, 9.618520025819415691344671914138, 10.73619653251474389820250689314, 11.26799399129559721959443888672

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