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Properties

Label 2-420-28.27-c1-0-26
Degree $2$
Conductor $420$
Sign $0.572 + 0.819i$
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  + (1.07 − 0.921i)2-s + 3-s + (0.303 − 1.97i)4-s + i·5-s + (1.07 − 0.921i)6-s + (1.82 + 1.91i)7-s + (−1.49 − 2.40i)8-s + 9-s + (0.921 + 1.07i)10-s − 6.24i·11-s + (0.303 − 1.97i)12-s + 2.40i·13-s + (3.72 + 0.373i)14-s + i·15-s + (−3.81 − 1.19i)16-s + 1.30i·17-s + ⋯
L(s)  = 1  + (0.758 − 0.651i)2-s + 0.577·3-s + (0.151 − 0.988i)4-s + 0.447i·5-s + (0.438 − 0.376i)6-s + (0.690 + 0.723i)7-s + (−0.528 − 0.848i)8-s + 0.333·9-s + (0.291 + 0.339i)10-s − 1.88i·11-s + (0.0875 − 0.570i)12-s + 0.666i·13-s + (0.995 + 0.0997i)14-s + 0.258i·15-s + (−0.954 − 0.299i)16-s + 0.317i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.572 + 0.819i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.572 + 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.572 + 0.819i$
Analytic conductor: \(3.35371\)
Root analytic conductor: \(1.83131\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{420} (391, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 420,\ (\ :1/2),\ 0.572 + 0.819i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.26346 - 1.18037i\)
\(L(\frac12)\) \(\approx\) \(2.26346 - 1.18037i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.07 + 0.921i)T \)
3 \( 1 - T \)
5 \( 1 - iT \)
7 \( 1 + (-1.82 - 1.91i)T \)
good11 \( 1 + 6.24iT - 11T^{2} \)
13 \( 1 - 2.40iT - 13T^{2} \)
17 \( 1 - 1.30iT - 17T^{2} \)
19 \( 1 - 3.94T + 19T^{2} \)
23 \( 1 - 3.55iT - 23T^{2} \)
29 \( 1 - 1.44T + 29T^{2} \)
31 \( 1 + 10.9T + 31T^{2} \)
37 \( 1 + 5.06T + 37T^{2} \)
41 \( 1 + 1.26iT - 41T^{2} \)
43 \( 1 - 2.19iT - 43T^{2} \)
47 \( 1 + 11.6T + 47T^{2} \)
53 \( 1 - 11.7T + 53T^{2} \)
59 \( 1 + 0.415T + 59T^{2} \)
61 \( 1 - 12.6iT - 61T^{2} \)
67 \( 1 - 3.10iT - 67T^{2} \)
71 \( 1 + 10.4iT - 71T^{2} \)
73 \( 1 - 1.64iT - 73T^{2} \)
79 \( 1 - 9.18iT - 79T^{2} \)
83 \( 1 + 7.39T + 83T^{2} \)
89 \( 1 + 11.4iT - 89T^{2} \)
97 \( 1 + 12.4iT - 97T^{2} \)
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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

−11.37459303272276891066082749435, −10.38945627715842376241486530626, −9.223731681836725482644125266416, −8.559031119915231216333663638446, −7.27837551313916817341513732510, −5.99987444884225613231568398708, −5.26788911534432267028286056287, −3.78560479649935341991665982326, −2.97974027380995383080739855081, −1.65570993276324022323700532366, 1.99539422190057795124183104029, 3.56918566197354991997716412404, 4.63091199771034515229478220894, 5.26517994419093800344363096029, 6.94448575583794824768324890866, 7.47937070847214439159704611486, 8.297837418335677517054758996631, 9.385589876792408710559873686748, 10.37492940559282501524889029180, 11.60144470860538183063310817714

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