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Properties

Label 1-3520-3520.1389-r0-0-0
Degree $1$
Conductor $3520$
Sign $0.731 - 0.681i$
Analytic cond. $16.3468$
Root an. cond. $16.3468$
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  + (0.233 + 0.972i)3-s + (−0.987 + 0.156i)7-s + (−0.891 + 0.453i)9-s + (0.649 − 0.760i)13-s + (0.951 − 0.309i)17-s + (−0.972 + 0.233i)19-s + (−0.382 − 0.923i)21-s + (−0.707 − 0.707i)23-s + (−0.649 − 0.760i)27-s + (−0.852 + 0.522i)29-s + (−0.309 + 0.951i)31-s + (0.972 + 0.233i)37-s + (0.891 + 0.453i)39-s + (−0.987 − 0.156i)41-s + (0.382 + 0.923i)43-s + ⋯
L(s)  = 1  + (0.233 + 0.972i)3-s + (−0.987 + 0.156i)7-s + (−0.891 + 0.453i)9-s + (0.649 − 0.760i)13-s + (0.951 − 0.309i)17-s + (−0.972 + 0.233i)19-s + (−0.382 − 0.923i)21-s + (−0.707 − 0.707i)23-s + (−0.649 − 0.760i)27-s + (−0.852 + 0.522i)29-s + (−0.309 + 0.951i)31-s + (0.972 + 0.233i)37-s + (0.891 + 0.453i)39-s + (−0.987 − 0.156i)41-s + (0.382 + 0.923i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.731 - 0.681i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.731 - 0.681i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3520\)    =    \(2^{6} \cdot 5 \cdot 11\)
Sign: $0.731 - 0.681i$
Analytic conductor: \(16.3468\)
Root analytic conductor: \(16.3468\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3520} (1389, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3520,\ (0:\ ),\ 0.731 - 0.681i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7857854624 - 0.3094273044i\)
\(L(\frac12)\) \(\approx\) \(0.7857854624 - 0.3094273044i\)
\(L(1)\) \(\approx\) \(0.8623987102 + 0.2086081318i\)
\(L(1)\) \(\approx\) \(0.8623987102 + 0.2086081318i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
11 \( 1 \)
good3 \( 1 + (0.233 + 0.972i)T \)
7 \( 1 + (-0.987 + 0.156i)T \)
13 \( 1 + (0.649 - 0.760i)T \)
17 \( 1 + (0.951 - 0.309i)T \)
19 \( 1 + (-0.972 + 0.233i)T \)
23 \( 1 + (-0.707 - 0.707i)T \)
29 \( 1 + (-0.852 + 0.522i)T \)
31 \( 1 + (-0.309 + 0.951i)T \)
37 \( 1 + (0.972 + 0.233i)T \)
41 \( 1 + (-0.987 - 0.156i)T \)
43 \( 1 + (0.382 + 0.923i)T \)
47 \( 1 + (0.587 + 0.809i)T \)
53 \( 1 + (-0.996 + 0.0784i)T \)
59 \( 1 + (0.972 + 0.233i)T \)
61 \( 1 + (-0.996 - 0.0784i)T \)
67 \( 1 + (0.382 - 0.923i)T \)
71 \( 1 + (-0.891 - 0.453i)T \)
73 \( 1 + (-0.156 - 0.987i)T \)
79 \( 1 + (-0.951 - 0.309i)T \)
83 \( 1 + (0.0784 - 0.996i)T \)
89 \( 1 + (0.707 - 0.707i)T \)
97 \( 1 + (0.309 - 0.951i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.82043349303213235799750405705, −18.45835799167751921050668970200, −17.31990641867786804666564769030, −16.91349635445779874897346499936, −16.17372050648206569585991657075, −15.31263052666391114397634000203, −14.58382677097005242474862388402, −13.82533267729289224575139941752, −13.20924499364825496512531250790, −12.78061590564761415878604876378, −11.89831660854450360657156582731, −11.38007563263137746772287262172, −10.38928304732928549062993834321, −9.57463491893715131081088754356, −8.95395253273330970805630216852, −8.13626663286920118274498636432, −7.4431932286594264902772224897, −6.70962298481126740417314747786, −6.06720885781874468278243035739, −5.53320566616724474038948710943, −3.99573065895782099067748616577, −3.65850147882585362284529443002, −2.5573406855850659129638765033, −1.87996215538979607225725127152, −0.89916843072512081655649355779, 0.27326623200209478588517445675, 1.701837486708057893702852819, 2.911991112536496014838712566176, 3.261989215063615961063709240017, 4.10078993987278685288328551203, 4.908515248526890644030515120129, 5.89366435130177604117909503556, 6.19365004704337466315488648817, 7.42142828123931844193418784236, 8.20939139953592531838933037520, 8.90261209685960032875146518408, 9.55306853792341397676212178750, 10.3304777916373861880979920226, 10.66309058091730519891209069926, 11.63027014786815972639434103173, 12.53399547800031063213958640834, 13.03426580412333715606079172203, 13.97379249241404368568326167605, 14.61428272970328955301717337876, 15.263491398807420946149927169413, 16.00393414034054574466416619875, 16.42711481988559376349424014677, 17.024955922782186830280665384558, 18.01962688382754732376427679755, 18.73509699648922784809396295544

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