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Properties

Label 2-420-105.89-c1-0-15
Degree $2$
Conductor $420$
Sign $-0.919 + 0.394i$
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  + (0.813 − 1.52i)3-s + (−1.61 − 1.55i)5-s + (−2.58 + 0.567i)7-s + (−1.67 − 2.48i)9-s + (0.793 − 0.457i)11-s − 4.31·13-s + (−3.68 + 1.20i)15-s + (0.465 − 0.268i)17-s + (−1.12 − 0.651i)19-s + (−1.23 + 4.41i)21-s + (3.99 − 6.91i)23-s + (0.193 + 4.99i)25-s + (−5.16 + 0.537i)27-s − 3.46i·29-s + (−5.56 + 3.21i)31-s + ⋯
L(s)  = 1  + (0.469 − 0.882i)3-s + (−0.720 − 0.693i)5-s + (−0.976 + 0.214i)7-s + (−0.558 − 0.829i)9-s + (0.239 − 0.138i)11-s − 1.19·13-s + (−0.950 + 0.310i)15-s + (0.112 − 0.0651i)17-s + (−0.258 − 0.149i)19-s + (−0.269 + 0.962i)21-s + (0.832 − 1.44i)23-s + (0.0386 + 0.999i)25-s + (−0.994 + 0.103i)27-s − 0.642i·29-s + (−0.998 + 0.576i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.919 + 0.394i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.919 + 0.394i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.919 + 0.394i$
Analytic conductor: \(3.35371\)
Root analytic conductor: \(1.83131\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{420} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 420,\ (\ :1/2),\ -0.919 + 0.394i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.167596 - 0.816114i\)
\(L(\frac12)\) \(\approx\) \(0.167596 - 0.816114i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.813 + 1.52i)T \)
5 \( 1 + (1.61 + 1.55i)T \)
7 \( 1 + (2.58 - 0.567i)T \)
good11 \( 1 + (-0.793 + 0.457i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 4.31T + 13T^{2} \)
17 \( 1 + (-0.465 + 0.268i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.12 + 0.651i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.99 + 6.91i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 3.46iT - 29T^{2} \)
31 \( 1 + (5.56 - 3.21i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.36 - 2.52i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 9.89T + 41T^{2} \)
43 \( 1 + 8.22iT - 43T^{2} \)
47 \( 1 + (3.75 + 2.17i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.984 - 1.70i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (7.15 + 12.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (8.38 + 4.84i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-10.4 + 6.05i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 0.943iT - 71T^{2} \)
73 \( 1 + (-5.11 - 8.85i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.71 + 9.90i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 9.35iT - 83T^{2} \)
89 \( 1 + (-0.874 + 1.51i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 10.7T + 97T^{2} \)
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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

−10.95550178974331055250611796156, −9.553063165460161534553445187363, −8.968602195132314234626077395887, −8.007816065078386466667016665279, −7.14150746583763601841953453357, −6.28970342397799752370971422313, −4.93300601409579097222173328391, −3.59382462512449936322370239968, −2.42593017078117974567554341737, −0.48430579865082527688088716467, 2.69139429687237224378641823735, 3.56065451825985014670011475941, 4.52313335295056899405178042647, 5.86419922584151741868186282566, 7.18895898803137909819578274331, 7.76145007601201575910862411828, 9.202399462509331166758465541720, 9.671984315558296960133506744091, 10.65281724543221153374153239937, 11.34640682602337073611571989190

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