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Properties

Label 1-3520-3520.3253-r0-0-0
Degree $1$
Conductor $3520$
Sign $0.516 + 0.856i$
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  + (−0.852 − 0.522i)3-s + (0.987 − 0.156i)7-s + (0.453 + 0.891i)9-s + (0.0784 + 0.996i)13-s + (0.309 + 0.951i)17-s + (−0.852 − 0.522i)19-s + (−0.923 − 0.382i)21-s + (−0.707 − 0.707i)23-s + (0.0784 − 0.996i)27-s + (0.972 + 0.233i)29-s + (−0.309 + 0.951i)31-s + (0.522 + 0.852i)37-s + (0.453 − 0.891i)39-s + (−0.156 + 0.987i)41-s + (0.382 − 0.923i)43-s + ⋯
L(s)  = 1  + (−0.852 − 0.522i)3-s + (0.987 − 0.156i)7-s + (0.453 + 0.891i)9-s + (0.0784 + 0.996i)13-s + (0.309 + 0.951i)17-s + (−0.852 − 0.522i)19-s + (−0.923 − 0.382i)21-s + (−0.707 − 0.707i)23-s + (0.0784 − 0.996i)27-s + (0.972 + 0.233i)29-s + (−0.309 + 0.951i)31-s + (0.522 + 0.852i)37-s + (0.453 − 0.891i)39-s + (−0.156 + 0.987i)41-s + (0.382 − 0.923i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.516 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.516 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3520\)    =    \(2^{6} \cdot 5 \cdot 11\)
Sign: $0.516 + 0.856i$
Analytic conductor: \(16.3468\)
Root analytic conductor: \(16.3468\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3520} (3253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3520,\ (0:\ ),\ 0.516 + 0.856i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.011725426 + 0.5711428501i\)
\(L(\frac12)\) \(\approx\) \(1.011725426 + 0.5711428501i\)
\(L(1)\) \(\approx\) \(0.8857476185 + 0.01104049226i\)
\(L(1)\) \(\approx\) \(0.8857476185 + 0.01104049226i\)

Euler product

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

−18.35837053899933838611275532456, −17.830983379930401788540947300765, −17.36197206358286148843692737906, −16.60221185836425911669446771316, −15.84080415906487935942542451518, −15.31190869357084965453297231284, −14.59816252755087389937366508029, −13.90824956091116263645922488608, −12.93249273425689703902856416454, −12.15119470994554424128478945771, −11.70972272389538806146186931434, −10.82051069283578455309877339688, −10.4782608442399698568486598434, −9.55961983977140710192846906164, −8.88927209027475525820826906680, −7.80460821403348711687923157721, −7.481748977870299129901957943166, −6.081396821673691205445366512686, −5.8471215001055517208358764985, −4.89364619474739399285533615245, −4.3729965715489745082615887858, −3.47875730475737280923654300544, −2.45279620045884008328563304844, −1.4134983675027844625638097421, −0.441492885663738150313160785654, 1.010313822017849931215137152952, 1.739590983205041220485852185300, 2.43025943655500547534001039334, 3.84673404933421095210816960658, 4.608922174727403828071142001406, 5.085479794180403593978128215637, 6.22139261521979184732192556385, 6.55473242773938264484844868086, 7.465163406623922985414753270341, 8.23648708267282077740119878277, 8.761994978990099758575763396, 9.96694744016981328964533938015, 10.72331310697606824013605982778, 11.0893110592964988979501096793, 12.06558209129802345588427529795, 12.32128706272752943829255175535, 13.32058748230293887935356328617, 13.984947853583717350455438397464, 14.61624722572773372063264917655, 15.44852859578357953174223008439, 16.32203461421425528743284106446, 16.948598215698612443333085605518, 17.3993416871456997320285404640, 18.1789916428056442236793544712, 18.66357081384280692142886644724

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