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Properties

Label 2-420-12.11-c1-0-47
Degree $2$
Conductor $420$
Sign $-0.953 + 0.300i$
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  + (1.09 − 0.899i)2-s + (−0.826 − 1.52i)3-s + (0.381 − 1.96i)4-s − i·5-s + (−2.27 − 0.917i)6-s − i·7-s + (−1.34 − 2.48i)8-s + (−1.63 + 2.51i)9-s + (−0.899 − 1.09i)10-s − 2.07·11-s + (−3.30 + 1.04i)12-s + 3.88·13-s + (−0.899 − 1.09i)14-s + (−1.52 + 0.826i)15-s + (−3.70 − 1.49i)16-s + 1.08i·17-s + ⋯
L(s)  = 1  + (0.771 − 0.636i)2-s + (−0.477 − 0.878i)3-s + (0.190 − 0.981i)4-s − 0.447i·5-s + (−0.927 − 0.374i)6-s − 0.377i·7-s + (−0.477 − 0.878i)8-s + (−0.544 + 0.838i)9-s + (−0.284 − 0.345i)10-s − 0.626·11-s + (−0.953 + 0.300i)12-s + 1.07·13-s + (−0.240 − 0.291i)14-s + (−0.393 + 0.213i)15-s + (−0.927 − 0.374i)16-s + 0.262i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.953 + 0.300i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.953 + 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.953 + 0.300i$
Analytic conductor: \(3.35371\)
Root analytic conductor: \(1.83131\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{420} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 420,\ (\ :1/2),\ -0.953 + 0.300i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.245526 - 1.59497i\)
\(L(\frac12)\) \(\approx\) \(0.245526 - 1.59497i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.09 + 0.899i)T \)
3 \( 1 + (0.826 + 1.52i)T \)
5 \( 1 + iT \)
7 \( 1 + iT \)
good11 \( 1 + 2.07T + 11T^{2} \)
13 \( 1 - 3.88T + 13T^{2} \)
17 \( 1 - 1.08iT - 17T^{2} \)
19 \( 1 + 1.74iT - 19T^{2} \)
23 \( 1 - 0.0176T + 23T^{2} \)
29 \( 1 - 9.28iT - 29T^{2} \)
31 \( 1 + 10.8iT - 31T^{2} \)
37 \( 1 - 6.46T + 37T^{2} \)
41 \( 1 + 9.66iT - 41T^{2} \)
43 \( 1 - 3.43iT - 43T^{2} \)
47 \( 1 - 0.213T + 47T^{2} \)
53 \( 1 + 7.93iT - 53T^{2} \)
59 \( 1 - 2.98T + 59T^{2} \)
61 \( 1 - 8.36T + 61T^{2} \)
67 \( 1 - 7.12iT - 67T^{2} \)
71 \( 1 - 4.44T + 71T^{2} \)
73 \( 1 - 5.62T + 73T^{2} \)
79 \( 1 + 2.33iT - 79T^{2} \)
83 \( 1 - 3.90T + 83T^{2} \)
89 \( 1 - 1.16iT - 89T^{2} \)
97 \( 1 - 11.4T + 97T^{2} \)
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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

−11.07811843583696864876702246823, −10.29202168322416457481149228225, −9.016572818064107480116229233379, −7.900766740198589463611578238179, −6.79061412893677416684649811920, −5.85743498397395734891324409319, −5.02752912594297242443066921017, −3.76358559752752895520637803499, −2.26066542351689864676509271519, −0.889897344191948568971815182299, 2.82309943488843072126203584704, 3.85168042324923775851481270297, 4.92550857187577857777627820478, 5.87745845200175683526628011526, 6.53764911855230012639240684657, 7.88322534710994584099365877436, 8.748733925955605358548213147061, 9.881511026222910225119017317604, 10.91233644039904461925269524414, 11.56228538983049126125662760768

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