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Properties

Label 2-420-105.2-c1-0-9
Degree $2$
Conductor $420$
Sign $0.598 + 0.801i$
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  + (−1.20 − 1.24i)3-s + (2.05 − 0.871i)5-s + (2.63 + 0.250i)7-s + (−0.112 + 2.99i)9-s + (1.95 − 1.12i)11-s + (−1.60 + 1.60i)13-s + (−3.56 − 1.52i)15-s + (4.12 + 1.10i)17-s + (−6.82 − 3.94i)19-s + (−2.85 − 3.58i)21-s + (8.63 − 2.31i)23-s + (3.48 − 3.58i)25-s + (3.87 − 3.46i)27-s − 3.34·29-s + (−2.63 − 4.55i)31-s + ⋯
L(s)  = 1  + (−0.693 − 0.720i)3-s + (0.920 − 0.389i)5-s + (0.995 + 0.0948i)7-s + (−0.0375 + 0.999i)9-s + (0.589 − 0.340i)11-s + (−0.446 + 0.446i)13-s + (−0.919 − 0.392i)15-s + (1.00 + 0.268i)17-s + (−1.56 − 0.904i)19-s + (−0.622 − 0.782i)21-s + (1.80 − 0.482i)23-s + (0.696 − 0.717i)25-s + (0.745 − 0.666i)27-s − 0.620·29-s + (−0.472 − 0.818i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.598 + 0.801i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.598 + 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.598 + 0.801i$
Analytic conductor: \(3.35371\)
Root analytic conductor: \(1.83131\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{420} (317, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 420,\ (\ :1/2),\ 0.598 + 0.801i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28049 - 0.641711i\)
\(L(\frac12)\) \(\approx\) \(1.28049 - 0.641711i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.20 + 1.24i)T \)
5 \( 1 + (-2.05 + 0.871i)T \)
7 \( 1 + (-2.63 - 0.250i)T \)
good11 \( 1 + (-1.95 + 1.12i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.60 - 1.60i)T - 13iT^{2} \)
17 \( 1 + (-4.12 - 1.10i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (6.82 + 3.94i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-8.63 + 2.31i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 3.34T + 29T^{2} \)
31 \( 1 + (2.63 + 4.55i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.41 + 0.379i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 5.07iT - 41T^{2} \)
43 \( 1 + (4.16 - 4.16i)T - 43iT^{2} \)
47 \( 1 + (3.26 + 12.1i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.11 - 4.15i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (0.733 + 1.26i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.02 - 6.96i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.91 - 10.8i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 9.99iT - 71T^{2} \)
73 \( 1 + (1.31 + 0.353i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (5.66 + 3.26i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.33 - 7.33i)T + 83iT^{2} \)
89 \( 1 + (-4.63 + 8.02i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.06 - 7.06i)T + 97iT^{2} \)
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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

−11.18004012700553867242795065807, −10.36212223739047602293881107473, −9.115880947322252240077138485779, −8.368616967421797548404744369135, −7.18077223211618699265013749973, −6.30698897530418123871802397987, −5.34352239321663378359353854745, −4.53853552890620202125686133360, −2.35696281838050254014670136654, −1.22102763534344823053646201199, 1.60281554651519043249053426302, 3.35945155847972143757184556281, 4.73891781939235289664404574163, 5.46335156037124747640288607469, 6.46001370839116176301171531449, 7.51302383919425270964047012288, 8.873532414177149614911471323574, 9.659053467980758943548713378065, 10.58647561710604339634328329753, 11.01609631818878676057484936550

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