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Properties

Label 2-420-7.4-c1-0-0
Degree $2$
Conductor $420$
Sign $-0.900 - 0.435i$
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  + (−0.5 + 0.866i)3-s + (0.5 + 0.866i)5-s + (−2.62 + 0.358i)7-s + (−0.499 − 0.866i)9-s + (−2.12 + 3.67i)11-s − 5.24·13-s − 0.999·15-s + (2.12 − 3.67i)17-s + (3.5 + 6.06i)19-s + (1 − 2.44i)21-s + (−2.12 − 3.67i)23-s + (−0.499 + 0.866i)25-s + 0.999·27-s − 10.2·29-s + (−3.74 + 6.48i)31-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (0.223 + 0.387i)5-s + (−0.990 + 0.135i)7-s + (−0.166 − 0.288i)9-s + (−0.639 + 1.10i)11-s − 1.45·13-s − 0.258·15-s + (0.514 − 0.891i)17-s + (0.802 + 1.39i)19-s + (0.218 − 0.534i)21-s + (−0.442 − 0.766i)23-s + (−0.0999 + 0.173i)25-s + 0.192·27-s − 1.90·29-s + (−0.672 + 1.16i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 - 0.435i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.900 - 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.900 - 0.435i$
Analytic conductor: \(3.35371\)
Root analytic conductor: \(1.83131\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{420} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 420,\ (\ :1/2),\ -0.900 - 0.435i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.134170 + 0.584907i\)
\(L(\frac12)\) \(\approx\) \(0.134170 + 0.584907i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (2.62 - 0.358i)T \)
good11 \( 1 + (2.12 - 3.67i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 5.24T + 13T^{2} \)
17 \( 1 + (-2.12 + 3.67i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.5 - 6.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.12 + 3.67i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 10.2T + 29T^{2} \)
31 \( 1 + (3.74 - 6.48i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.62 - 4.54i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 4.24T + 41T^{2} \)
43 \( 1 + 5.24T + 43T^{2} \)
47 \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.24 + 7.34i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.878 + 1.52i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.24 - 10.8i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.62 - 2.80i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12.7T + 71T^{2} \)
73 \( 1 + (0.378 - 0.655i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.5 + 9.52i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 1.75T + 83T^{2} \)
89 \( 1 + (-0.878 - 1.52i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 16.4T + 97T^{2} \)
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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

−11.69000904294307145671955618731, −10.26224856429890085227336501650, −9.969322736459634117230620230834, −9.268523245521870276845678568423, −7.65034272377223948055316268734, −7.01852940299237952409717322203, −5.74860372733934330100202842154, −4.94357592168906617971121031521, −3.55013538215165469220858220608, −2.38530391776947463496428850849, 0.37023026769268605719783216471, 2.37443482071411292032872693957, 3.66946469448207404319527550891, 5.31768711208281404112084047763, 5.88716998267454923762085605991, 7.17193146483779123811676426189, 7.83347337636037704019412163322, 9.173698615374109493697900508881, 9.782022004739855902464720006448, 10.89901741914282003128731771346

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