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Properties

Label 1-3520-3520.1589-r0-0-0
Degree $1$
Conductor $3520$
Sign $0.978 + 0.207i$
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  + (−0.649 − 0.760i)3-s + (−0.891 + 0.453i)7-s + (−0.156 + 0.987i)9-s + (−0.852 − 0.522i)13-s + (0.587 − 0.809i)17-s + (−0.760 + 0.649i)19-s + (0.923 + 0.382i)21-s + (0.707 − 0.707i)23-s + (0.852 − 0.522i)27-s + (0.0784 + 0.996i)29-s + (0.809 − 0.587i)31-s + (0.760 + 0.649i)37-s + (0.156 + 0.987i)39-s + (−0.891 − 0.453i)41-s + (−0.923 − 0.382i)43-s + ⋯
L(s)  = 1  + (−0.649 − 0.760i)3-s + (−0.891 + 0.453i)7-s + (−0.156 + 0.987i)9-s + (−0.852 − 0.522i)13-s + (0.587 − 0.809i)17-s + (−0.760 + 0.649i)19-s + (0.923 + 0.382i)21-s + (0.707 − 0.707i)23-s + (0.852 − 0.522i)27-s + (0.0784 + 0.996i)29-s + (0.809 − 0.587i)31-s + (0.760 + 0.649i)37-s + (0.156 + 0.987i)39-s + (−0.891 − 0.453i)41-s + (−0.923 − 0.382i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.207i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.207i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3520\)    =    \(2^{6} \cdot 5 \cdot 11\)
Sign: $0.978 + 0.207i$
Analytic conductor: \(16.3468\)
Root analytic conductor: \(16.3468\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3520} (1589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3520,\ (0:\ ),\ 0.978 + 0.207i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7159274365 + 0.07526700954i\)
\(L(\frac12)\) \(\approx\) \(0.7159274365 + 0.07526700954i\)
\(L(1)\) \(\approx\) \(0.6794845246 - 0.1118782862i\)
\(L(1)\) \(\approx\) \(0.6794845246 - 0.1118782862i\)

Euler product

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

−18.835830622325839178665878355395, −17.76320421334944717776504994909, −17.13258207556091789029535509538, −16.75818980062003058130013849776, −16.08078138869327200975284689953, −15.24542393649734408104430564644, −14.85201848242287894034684925002, −13.893148337748214462199673870046, −13.03881168844717306444099211231, −12.47389932584977171352192452131, −11.636131924535856203342570652595, −11.05770652316478662635866572020, −10.09715247815936137829226514110, −9.85786430949310101412731868732, −9.08515287775805539476085693582, −8.17912057567706315357206431582, −7.13072430859822867448143429053, −6.5150625822102256819083632899, −5.905078800174091253960413436048, −4.89350078961978131899595322780, −4.37639947617319605991906096153, −3.49332709245890353168983643375, −2.84503945695532241225342012301, −1.56815892663398161124815507875, −0.37339836045416506076237909908, 0.62126531714991553703239473858, 1.701059777503054999634634611235, 2.6911726386440358754759042917, 3.18300089376386485506107841254, 4.57248769305667040898674451248, 5.20123595035996151496215455709, 5.99264681247952777601405496330, 6.62466602278413676254050887751, 7.26348424284528233728529485525, 8.07642491788118752617159921726, 8.806396987925490355311288366093, 9.86641241745456982302328297480, 10.2660979558560152667037915462, 11.22091333549880109792693421280, 12.02599511585480451396741090376, 12.464559947303586238315700771860, 13.039124272726573546756997583481, 13.75464610401737448257109407111, 14.67208735062474883594753268068, 15.2915524179873345288943386371, 16.29648716408621502534959107606, 16.70540009256589928392182534468, 17.31003746486658785278641389268, 18.264258285927894331581770617186, 18.68476141185898543689556791432

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