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Properties

Label 2-420-60.59-c1-0-7
Degree $2$
Conductor $420$
Sign $0.507 - 0.861i$
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  + (−0.835 − 1.14i)2-s + (−1.10 + 1.33i)3-s + (−0.602 + 1.90i)4-s + (−1 − 2i)5-s + (2.44 + 0.140i)6-s − 7-s + (2.67 − 0.907i)8-s + (−0.569 − 2.94i)9-s + (−1.44 + 2.81i)10-s + 2.20·11-s + (−1.88 − 2.90i)12-s + 1.89i·13-s + (0.835 + 1.14i)14-s + (3.77 + 0.868i)15-s + (−3.27 − 2.29i)16-s − 1.13·17-s + ⋯
L(s)  = 1  + (−0.591 − 0.806i)2-s + (−0.636 + 0.771i)3-s + (−0.301 + 0.953i)4-s + (−0.447 − 0.894i)5-s + (0.998 + 0.0574i)6-s − 0.377·7-s + (0.947 − 0.320i)8-s + (−0.189 − 0.981i)9-s + (−0.457 + 0.889i)10-s + 0.664·11-s + (−0.543 − 0.839i)12-s + 0.524i·13-s + (0.223 + 0.304i)14-s + (0.974 + 0.224i)15-s + (−0.818 − 0.574i)16-s − 0.276·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.507 - 0.861i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.507 - 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.507 - 0.861i$
Analytic conductor: \(3.35371\)
Root analytic conductor: \(1.83131\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{420} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 420,\ (\ :1/2),\ 0.507 - 0.861i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.442094 + 0.252716i\)
\(L(\frac12)\) \(\approx\) \(0.442094 + 0.252716i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.835 + 1.14i)T \)
3 \( 1 + (1.10 - 1.33i)T \)
5 \( 1 + (1 + 2i)T \)
7 \( 1 + T \)
good11 \( 1 - 2.20T + 11T^{2} \)
13 \( 1 - 1.89iT - 13T^{2} \)
17 \( 1 + 1.13T + 17T^{2} \)
19 \( 1 - 8.56iT - 19T^{2} \)
23 \( 1 - 3.21iT - 23T^{2} \)
29 \( 1 - 1.89iT - 29T^{2} \)
31 \( 1 - 5.90iT - 31T^{2} \)
37 \( 1 + 0.409iT - 37T^{2} \)
41 \( 1 - 4.40iT - 41T^{2} \)
43 \( 1 - 0.934T + 43T^{2} \)
47 \( 1 + 2.67iT - 47T^{2} \)
53 \( 1 - 6.81T + 53T^{2} \)
59 \( 1 + 13.5T + 59T^{2} \)
61 \( 1 - 12.4T + 61T^{2} \)
67 \( 1 - 10.8T + 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + 8iT - 73T^{2} \)
79 \( 1 + 3.76iT - 79T^{2} \)
83 \( 1 - 6.84iT - 83T^{2} \)
89 \( 1 - 16.1iT - 89T^{2} \)
97 \( 1 - 18.4iT - 97T^{2} \)
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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

−11.36966071279486340727233721775, −10.40370789750666402563430731322, −9.571962703060480842230661264155, −8.961953685616662702822692612427, −8.017740951710608251769645526049, −6.69293154969909903357677331332, −5.35412191310073284077441711698, −4.17519326988954666241878686399, −3.54239195569260848176323692745, −1.36795601593671980633127721918, 0.47369387241390498603960968426, 2.49528189785258032153258981217, 4.40844069502692363255076583349, 5.72396896746588193545403955920, 6.64965698631194128647019898698, 7.09320591655988701519493107969, 8.043399370359321171061612036133, 9.078231518349533071607321705403, 10.21280591831085816435345759767, 11.04003805269431865641757428630

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