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Properties

Label 1-3503-3503.51-r0-0-0
Degree $1$
Conductor $3503$
Sign $-0.0669 + 0.997i$
Analytic cond. $16.2678$
Root an. cond. $16.2678$
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  + (0.963 + 0.266i)2-s + (−0.427 + 0.904i)3-s + (0.858 + 0.512i)4-s + (0.185 − 0.982i)5-s + (−0.652 + 0.757i)6-s + (0.791 − 0.611i)7-s + (0.691 + 0.722i)8-s + (−0.635 − 0.772i)9-s + (0.440 − 0.897i)10-s + (0.817 + 0.575i)11-s + (−0.830 + 0.557i)12-s + (0.762 + 0.646i)13-s + (0.925 − 0.379i)14-s + (0.809 + 0.587i)15-s + (0.473 + 0.880i)16-s + (0.663 + 0.748i)17-s + ⋯
L(s)  = 1  + (0.963 + 0.266i)2-s + (−0.427 + 0.904i)3-s + (0.858 + 0.512i)4-s + (0.185 − 0.982i)5-s + (−0.652 + 0.757i)6-s + (0.791 − 0.611i)7-s + (0.691 + 0.722i)8-s + (−0.635 − 0.772i)9-s + (0.440 − 0.897i)10-s + (0.817 + 0.575i)11-s + (−0.830 + 0.557i)12-s + (0.762 + 0.646i)13-s + (0.925 − 0.379i)14-s + (0.809 + 0.587i)15-s + (0.473 + 0.880i)16-s + (0.663 + 0.748i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3503 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0669 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3503 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0669 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3503\)    =    \(31 \cdot 113\)
Sign: $-0.0669 + 0.997i$
Analytic conductor: \(16.2678\)
Root analytic conductor: \(16.2678\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3503} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3503,\ (0:\ ),\ -0.0669 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.426325773 + 2.594530036i\)
\(L(\frac12)\) \(\approx\) \(2.426325773 + 2.594530036i\)
\(L(1)\) \(\approx\) \(1.815124397 + 0.8391191778i\)
\(L(1)\) \(\approx\) \(1.815124397 + 0.8391191778i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad31 \( 1 \)
113 \( 1 \)
good2 \( 1 + (0.963 + 0.266i)T \)
3 \( 1 + (-0.427 + 0.904i)T \)
5 \( 1 + (0.185 - 0.982i)T \)
7 \( 1 + (0.791 - 0.611i)T \)
11 \( 1 + (0.817 + 0.575i)T \)
13 \( 1 + (0.762 + 0.646i)T \)
17 \( 1 + (0.663 + 0.748i)T \)
19 \( 1 + (-0.685 + 0.727i)T \)
23 \( 1 + (-0.795 + 0.605i)T \)
29 \( 1 + (0.493 + 0.869i)T \)
37 \( 1 + (-0.0373 + 0.999i)T \)
41 \( 1 + (0.119 - 0.992i)T \)
43 \( 1 + (-0.862 - 0.506i)T \)
47 \( 1 + (-0.0672 + 0.997i)T \)
53 \( 1 + (-0.420 + 0.907i)T \)
59 \( 1 + (0.904 + 0.427i)T \)
61 \( 1 + (-0.974 + 0.222i)T \)
67 \( 1 + (0.467 + 0.884i)T \)
71 \( 1 + (-0.544 - 0.838i)T \)
73 \( 1 + (-0.998 - 0.0523i)T \)
79 \( 1 + (0.171 - 0.985i)T \)
83 \( 1 + (0.712 - 0.701i)T \)
89 \( 1 + (0.200 - 0.979i)T \)
97 \( 1 + (-0.691 + 0.722i)T \)
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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

−18.50258706251415926567568323624, −18.115163332495771188583385860697, −17.333638689861374283257909394655, −16.42218058081166082554014678266, −15.66870250892535964276860963101, −14.77484775808511887694291327075, −14.367969821454597960336706054202, −13.70344282302303510594068350414, −13.135678058802087962596941836659, −12.18317545778626296517819954975, −11.65664708040506718079458147161, −11.15684232626152885794672060015, −10.61269966169614307342097898110, −9.63273875426273214949831809889, −8.37459969257132533660845136182, −7.83861457978350634132720686346, −6.81122066717585025443656035894, −6.358478329724210698194598550565, −5.74603532415258039734462942681, −5.07397966940138081809796770439, −4.02966647706400972674211779351, −3.07607062758246620148440929948, −2.436301885459221682309734431447, −1.72287433834940656442568401266, −0.75947798796736992720743619011, 1.390226869803078812020995090194, 1.71836340678709235575224073006, 3.338693520587052417100871318643, 4.15187764890289295639681345287, 4.30244216469377593204625773909, 5.159986575298434764283031822005, 5.90406408970107706549320454100, 6.464685037799237761711288914909, 7.5296514058341914827106121773, 8.397829037898887953481990462598, 8.93035864725541255555731215095, 10.05785369812107865365684009024, 10.56455357481538392381647435014, 11.49551917100825366311517730510, 12.01336635551424855666186045979, 12.536378269827707235167868321859, 13.5898013849325364235977226227, 14.15349399236591558286544707069, 14.76079007844889400159081801474, 15.4231661045403253333636769290, 16.41819644513934634216046394421, 16.52583764035367660673714311399, 17.38796014301076206985246774033, 17.66170860194715279505720649206, 19.13805275547659568152722034304

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