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Properties

Label 2-420-28.3-c1-0-4
Degree $2$
Conductor $420$
Sign $0.895 - 0.444i$
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 − i)2-s + (0.5 − 0.866i)3-s + 2i·4-s + (−0.866 + 0.5i)5-s + (−1.36 + 0.366i)6-s + (−0.5 + 2.59i)7-s + (2 − 2i)8-s + (−0.499 − 0.866i)9-s + (1.36 + 0.366i)10-s + (−1.73 − i)11-s + (1.73 + i)12-s + 3.73i·13-s + (3.09 − 2.09i)14-s + 0.999i·15-s − 4·16-s + (6 + 3.46i)17-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + (0.288 − 0.499i)3-s + i·4-s + (−0.387 + 0.223i)5-s + (−0.557 + 0.149i)6-s + (−0.188 + 0.981i)7-s + (0.707 − 0.707i)8-s + (−0.166 − 0.288i)9-s + (0.431 + 0.115i)10-s + (−0.522 − 0.301i)11-s + (0.499 + 0.288i)12-s + 1.03i·13-s + (0.827 − 0.560i)14-s + 0.258i·15-s − 16-s + (1.45 + 0.840i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.895 - 0.444i$
Analytic conductor: \(3.35371\)
Root analytic conductor: \(1.83131\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{420} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 420,\ (\ :1/2),\ 0.895 - 0.444i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.819176 + 0.191901i\)
\(L(\frac12)\) \(\approx\) \(0.819176 + 0.191901i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (0.5 - 2.59i)T \)
good11 \( 1 + (1.73 + i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 3.73iT - 13T^{2} \)
17 \( 1 + (-6 - 3.46i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.23 - 3.86i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (6.92 - 4i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 9.46T + 29T^{2} \)
31 \( 1 + (0.232 - 0.401i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.23 - 3.86i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 6iT - 41T^{2} \)
43 \( 1 - 1.19iT - 43T^{2} \)
47 \( 1 + (3.73 + 6.46i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.73 - 8.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.46 + 9.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.53 - 1.46i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.96 - 4.59i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 8.53iT - 71T^{2} \)
73 \( 1 + (-6.23 - 3.59i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.69 - 3.86i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 2T + 83T^{2} \)
89 \( 1 + (3.80 - 2.19i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 1.07iT - 97T^{2} \)
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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

−11.44953446813232900991074946956, −10.18328510440999524268205715778, −9.584601823712575163878147912917, −8.267251887912984021940851665613, −8.097736411986031471853808065656, −6.80822835808636887413494411107, −5.66232878047300777304000345770, −3.88682653668595991178059538036, −2.89111900236793762470330141692, −1.62911439397576681304264936851, 0.69344408710602790532978639240, 2.95882902574004375627074912277, 4.47458405778873887908822078185, 5.33331584176933187633913425429, 6.64009052540507893477019120956, 7.85122310648986188901733607922, 7.999341954034409427258602619972, 9.428413124558293897468069195187, 10.10833405064699669981082128932, 10.66011660218324077693177869304

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