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Properties

Label 2-420-140.139-c1-0-46
Degree $2$
Conductor $420$
Sign $-0.982 + 0.183i$
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  + (0.654 − 1.25i)2-s − i·3-s + (−1.14 − 1.64i)4-s + (0.206 − 2.22i)5-s + (−1.25 − 0.654i)6-s + (0.859 − 2.50i)7-s + (−2.80 + 0.359i)8-s − 9-s + (−2.65 − 1.71i)10-s + 4.14i·11-s + (−1.64 + 1.14i)12-s + 3.35·13-s + (−2.57 − 2.71i)14-s + (−2.22 − 0.206i)15-s + (−1.38 + 3.75i)16-s + 3.18·17-s + ⋯
L(s)  = 1  + (0.462 − 0.886i)2-s − 0.577i·3-s + (−0.571 − 0.820i)4-s + (0.0925 − 0.995i)5-s + (−0.511 − 0.267i)6-s + (0.324 − 0.945i)7-s + (−0.991 + 0.127i)8-s − 0.333·9-s + (−0.839 − 0.542i)10-s + 1.24i·11-s + (−0.473 + 0.330i)12-s + 0.930·13-s + (−0.688 − 0.725i)14-s + (−0.574 − 0.0534i)15-s + (−0.346 + 0.938i)16-s + 0.773·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 + 0.183i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.982 + 0.183i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.982 + 0.183i$
Analytic conductor: \(3.35371\)
Root analytic conductor: \(1.83131\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{420} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 420,\ (\ :1/2),\ -0.982 + 0.183i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.148718 - 1.60279i\)
\(L(\frac12)\) \(\approx\) \(0.148718 - 1.60279i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.654 + 1.25i)T \)
3 \( 1 + iT \)
5 \( 1 + (-0.206 + 2.22i)T \)
7 \( 1 + (-0.859 + 2.50i)T \)
good11 \( 1 - 4.14iT - 11T^{2} \)
13 \( 1 - 3.35T + 13T^{2} \)
17 \( 1 - 3.18T + 17T^{2} \)
19 \( 1 + 6.52T + 19T^{2} \)
23 \( 1 - 1.42T + 23T^{2} \)
29 \( 1 - 8.88T + 29T^{2} \)
31 \( 1 + 2.93T + 31T^{2} \)
37 \( 1 + 9.54iT - 37T^{2} \)
41 \( 1 + 3.55iT - 41T^{2} \)
43 \( 1 - 0.667T + 43T^{2} \)
47 \( 1 + 0.693iT - 47T^{2} \)
53 \( 1 - 4.18iT - 53T^{2} \)
59 \( 1 - 5.96T + 59T^{2} \)
61 \( 1 - 10.3iT - 61T^{2} \)
67 \( 1 - 7.05T + 67T^{2} \)
71 \( 1 - 9.50iT - 71T^{2} \)
73 \( 1 - 7.52T + 73T^{2} \)
79 \( 1 + 12.1iT - 79T^{2} \)
83 \( 1 + 5.25iT - 83T^{2} \)
89 \( 1 - 2.83iT - 89T^{2} \)
97 \( 1 + 15.9T + 97T^{2} \)
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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

−10.79919366381186919036625495549, −10.12010665244638964894190931524, −9.010466588199799068330111751159, −8.188287804481274657312106444542, −6.96464578622825772326788271723, −5.77238259942165452348404338760, −4.65010158590550242300211775958, −3.89630944895596953248850354342, −2.07578436457102846248771321365, −0.971186449491268218642603356052, 2.82454722376829563423381684812, 3.69505680951774112242413359861, 5.05466917482082882367169182858, 6.07498353273040888382516403911, 6.54943367869238442848388940475, 8.234209493276117905948785980831, 8.475507822222692819877140571681, 9.709556096103760267870573765034, 10.86209725699262356691027899433, 11.51513556937824138034447409784

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