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Properties

Label 2-420-28.19-c1-0-26
Degree $2$
Conductor $420$
Sign $0.961 - 0.275i$
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 − 0.0446i)2-s + (0.5 + 0.866i)3-s + (1.99 − 0.126i)4-s + (0.866 + 0.5i)5-s + (0.745 + 1.20i)6-s + (1.71 − 2.01i)7-s + (2.81 − 0.267i)8-s + (−0.499 + 0.866i)9-s + (1.24 + 0.668i)10-s + (−4.26 + 2.46i)11-s + (1.10 + 1.66i)12-s − 6.68i·13-s + (2.33 − 2.92i)14-s + 0.999i·15-s + (3.96 − 0.503i)16-s + (−2.69 + 1.55i)17-s + ⋯
L(s)  = 1  + (0.999 − 0.0315i)2-s + (0.288 + 0.499i)3-s + (0.998 − 0.0631i)4-s + (0.387 + 0.223i)5-s + (0.304 + 0.490i)6-s + (0.647 − 0.761i)7-s + (0.995 − 0.0945i)8-s + (−0.166 + 0.288i)9-s + (0.394 + 0.211i)10-s + (−1.28 + 0.741i)11-s + (0.319 + 0.480i)12-s − 1.85i·13-s + (0.623 − 0.781i)14-s + 0.258i·15-s + (0.992 − 0.125i)16-s + (−0.653 + 0.377i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 - 0.275i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.961 - 0.275i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.961 - 0.275i$
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: \((2,\ 420,\ (\ :1/2),\ 0.961 - 0.275i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.86095 + 0.401276i\)
\(L(\frac12)\) \(\approx\) \(2.86095 + 0.401276i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 + 0.0446i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (-1.71 + 2.01i)T \)
good11 \( 1 + (4.26 - 2.46i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 6.68iT - 13T^{2} \)
17 \( 1 + (2.69 - 1.55i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.00 - 6.93i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.821 - 0.474i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 3.42T + 29T^{2} \)
31 \( 1 + (2.92 + 5.07i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.100 - 0.174i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 3.14iT - 41T^{2} \)
43 \( 1 - 6.69iT - 43T^{2} \)
47 \( 1 + (3.47 - 6.02i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.11 - 3.66i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.59 + 6.22i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (11.6 + 6.75i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.17 + 1.83i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 7.61iT - 71T^{2} \)
73 \( 1 + (-3.69 + 2.13i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.52 - 2.03i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 3.03T + 83T^{2} \)
89 \( 1 + (-15.1 - 8.77i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 7.42iT - 97T^{2} \)
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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

−10.84227357273795311264657291104, −10.64316342446607710934881094527, −9.888511022284958479011771247558, −7.980729613273520110998348816966, −7.76604238032715676916697118278, −6.24355632802046583931823170116, −5.27400324586628640857554365286, −4.42371876359742954429484756957, −3.25163458271594768572044126084, −2.04237851866740251439108045633, 1.97602014821543699157708934758, 2.75610431768257945008358090248, 4.50024353805442981481456348029, 5.25479674433451240734320936581, 6.40515553640624326225737483773, 7.15359510110521308062054632028, 8.466197426503269988973091842703, 9.036582085619698682262091706506, 10.64907743459245684600505830949, 11.38160440929221147312372668538

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