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Properties

Label 1-3520-3520.1549-r0-0-0
Degree $1$
Conductor $3520$
Sign $-0.544 + 0.838i$
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  + (0.760 + 0.649i)3-s + (0.891 + 0.453i)7-s + (0.156 + 0.987i)9-s + (0.522 + 0.852i)13-s + (0.587 + 0.809i)17-s + (−0.649 + 0.760i)19-s + (0.382 + 0.923i)21-s + (−0.707 − 0.707i)23-s + (−0.522 + 0.852i)27-s + (−0.996 − 0.0784i)29-s + (0.809 + 0.587i)31-s + (0.649 + 0.760i)37-s + (−0.156 + 0.987i)39-s + (0.891 − 0.453i)41-s + (−0.382 − 0.923i)43-s + ⋯
L(s)  = 1  + (0.760 + 0.649i)3-s + (0.891 + 0.453i)7-s + (0.156 + 0.987i)9-s + (0.522 + 0.852i)13-s + (0.587 + 0.809i)17-s + (−0.649 + 0.760i)19-s + (0.382 + 0.923i)21-s + (−0.707 − 0.707i)23-s + (−0.522 + 0.852i)27-s + (−0.996 − 0.0784i)29-s + (0.809 + 0.587i)31-s + (0.649 + 0.760i)37-s + (−0.156 + 0.987i)39-s + (0.891 − 0.453i)41-s + (−0.382 − 0.923i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.544 + 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.544 + 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3520\)    =    \(2^{6} \cdot 5 \cdot 11\)
Sign: $-0.544 + 0.838i$
Analytic conductor: \(16.3468\)
Root analytic conductor: \(16.3468\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3520} (1549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3520,\ (0:\ ),\ -0.544 + 0.838i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.187785427 + 2.187484453i\)
\(L(\frac12)\) \(\approx\) \(1.187785427 + 2.187484453i\)
\(L(1)\) \(\approx\) \(1.340091280 + 0.6891599016i\)
\(L(1)\) \(\approx\) \(1.340091280 + 0.6891599016i\)

Euler product

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

−18.49292519503216620561877632708, −17.77889604239563581033250510784, −17.485603566316879821976608705473, −16.417513607281049065557234864308, −15.61561577290565894804750572380, −14.85856003697067894959693592290, −14.409621671348924236500081413737, −13.56160056290171305226005271326, −13.1788800697101707283872577935, −12.37028891322429815982956764040, −11.4580755272061804563969773271, −11.0188760159957645394947324579, −9.92878460283972619642328905095, −9.33586627795233700833545936462, −8.42490507234795432947732760526, −7.79425838861654056814414360390, −7.461305875948683911183123747196, −6.431373326385708850307679144, −5.69313258319248301887070666185, −4.71228260943208645906307243851, −3.910303603392728430882265306250, −3.090324031288618874038975914, −2.29261475606916973833844241907, −1.41810297898147905470245087650, −0.628725759607163139864221692289, 1.46518896990730685504389620748, 2.0061389077021859857714697010, 2.88856767148822125801647875917, 4.03922139134784065107363777151, 4.203211485329441132675479620723, 5.30955904977124200084682490672, 5.96480388296448791180601639685, 6.97328522397604152118641340185, 8.01586559380702829235081195323, 8.39589192721703244154844913958, 8.95314717438726036773614480600, 9.9550110830029179609177451960, 10.42700535887417779582725673837, 11.284943058260918207006163162238, 11.924437640919539105136894959107, 12.80932764234956211689518979708, 13.61332321019537724684067452019, 14.414674409566735243909139844257, 14.69878989562082498310533672553, 15.384938684111154945198502475264, 16.27971222896580254036398058407, 16.70263916121989613040928841390, 17.57632715571334353236478687060, 18.51816526302576048795606081697, 18.92365293988944096904611504681

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