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Properties

Label 2-420-420.299-c0-0-1
Degree $2$
Conductor $420$
Sign $0.605 - 0.795i$
Analytic cond. $0.209607$
Root an. cond. $0.457828$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 + 0.5i)2-s + (−0.866 − 0.5i)3-s + (0.499 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (−0.499 − 0.866i)6-s + (0.866 − 0.5i)7-s + 0.999i·8-s + (0.499 + 0.866i)9-s + (−0.866 + 0.499i)10-s − 0.999i·12-s + 0.999·14-s + (0.866 − 0.499i)15-s + (−0.5 + 0.866i)16-s + 0.999i·18-s − 0.999·20-s − 0.999·21-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (−0.866 − 0.5i)3-s + (0.499 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (−0.499 − 0.866i)6-s + (0.866 − 0.5i)7-s + 0.999i·8-s + (0.499 + 0.866i)9-s + (−0.866 + 0.499i)10-s − 0.999i·12-s + 0.999·14-s + (0.866 − 0.499i)15-s + (−0.5 + 0.866i)16-s + 0.999i·18-s − 0.999·20-s − 0.999·21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.605 - 0.795i$
Analytic conductor: \(0.209607\)
Root analytic conductor: \(0.457828\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{420} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 420,\ (\ :0),\ 0.605 - 0.795i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.039953995\)
\(L(\frac12)\) \(\approx\) \(1.039953995\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-0.866 + 0.5i)T \)
good11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 + 1.73iT - T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 - T + T^{2} \)
43 \( 1 + 1.73T + T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 - 1.73T + T^{2} \)
89 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + T^{2} \)
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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

−11.62366313164651672672926009597, −11.04318473098522800255480370603, −10.16848482126727609339755252435, −8.146432072520953842454392025104, −7.67715071722314271315318538649, −6.74022205291033284051633369315, −5.98922457116934028701428797533, −4.79170469345452232999176548548, −3.91739845828293122628002374472, −2.25701509693477746751512627437, 1.54593828982779441845747508647, 3.54166832869921882151477494394, 4.63348731982615795017550352544, 5.19356221722529463781906838057, 6.10599717856655134472937805802, 7.43953299393658909202255084721, 8.748509513461255698812594491069, 9.690261493512842408992541248743, 10.76907287115124302420922169815, 11.42493748981397774228058826447

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