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Properties

Label 2-420-28.19-c1-0-14
Degree $2$
Conductor $420$
Sign $0.759 + 0.650i$
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  + (−1.41 + 0.0654i)2-s + (−0.5 − 0.866i)3-s + (1.99 − 0.185i)4-s + (−0.866 − 0.5i)5-s + (0.763 + 1.19i)6-s + (2.61 + 0.395i)7-s + (−2.80 + 0.391i)8-s + (−0.499 + 0.866i)9-s + (1.25 + 0.649i)10-s + (2.76 − 1.59i)11-s + (−1.15 − 1.63i)12-s + 4.22i·13-s + (−3.72 − 0.386i)14-s + 0.999i·15-s + (3.93 − 0.737i)16-s + (0.0240 − 0.0138i)17-s + ⋯
L(s)  = 1  + (−0.998 + 0.0463i)2-s + (−0.288 − 0.499i)3-s + (0.995 − 0.0925i)4-s + (−0.387 − 0.223i)5-s + (0.311 + 0.486i)6-s + (0.988 + 0.149i)7-s + (−0.990 + 0.138i)8-s + (−0.166 + 0.288i)9-s + (0.397 + 0.205i)10-s + (0.834 − 0.482i)11-s + (−0.333 − 0.471i)12-s + 1.17i·13-s + (−0.994 − 0.103i)14-s + 0.258i·15-s + (0.982 − 0.184i)16-s + (0.00583 − 0.00336i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.759 + 0.650i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.759 + 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.759 + 0.650i$
Analytic conductor: \(3.35371\)
Root analytic conductor: \(1.83131\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{420} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 420,\ (\ :1/2),\ 0.759 + 0.650i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.822522 - 0.303786i\)
\(L(\frac12)\) \(\approx\) \(0.822522 - 0.303786i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 - 0.0654i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 + (-2.61 - 0.395i)T \)
good11 \( 1 + (-2.76 + 1.59i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 4.22iT - 13T^{2} \)
17 \( 1 + (-0.0240 + 0.0138i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.21 + 3.83i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.14 + 1.81i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 9.93T + 29T^{2} \)
31 \( 1 + (0.507 + 0.878i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.61 + 4.52i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 7.14iT - 41T^{2} \)
43 \( 1 + 8.40iT - 43T^{2} \)
47 \( 1 + (-1.94 + 3.36i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.74 - 4.74i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.39 - 7.61i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.651 - 0.376i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-14.0 + 8.11i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 9.83iT - 71T^{2} \)
73 \( 1 + (13.1 - 7.58i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (12.3 + 7.11i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 9.71T + 83T^{2} \)
89 \( 1 + (8.85 + 5.11i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 12.1iT - 97T^{2} \)
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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

−11.28712652381430862248624502019, −10.23031553884219848929255287074, −8.896753538563767848168137377455, −8.578289341865067821872685829024, −7.40490743953456857053317500783, −6.71903335183422438656161304393, −5.57910046451889144807904135344, −4.19161318037360083169508583065, −2.33077925108252433536028591738, −1.01086571199706272489347723155, 1.27521041201946314466857712324, 3.05756379194334702990647451582, 4.38884029262472710536585799514, 5.70045599567122582128289113104, 6.79929130998981118096150098619, 7.932152308276409076836565390754, 8.379961081093098527115812371165, 9.764470797228378780785924234823, 10.21586183748460644915242204511, 11.25588597589005312433063508099

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