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Properties

Label 1-3520-3520.1467-r0-0-0
Degree $1$
Conductor $3520$
Sign $0.964 - 0.263i$
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  + (0.972 + 0.233i)3-s + (0.156 − 0.987i)7-s + (0.891 + 0.453i)9-s + (0.760 − 0.649i)13-s + (0.309 − 0.951i)17-s + (0.972 + 0.233i)19-s + (0.382 − 0.923i)21-s + (0.707 + 0.707i)23-s + (0.760 + 0.649i)27-s + (−0.852 − 0.522i)29-s + (0.309 + 0.951i)31-s + (0.233 + 0.972i)37-s + (0.891 − 0.453i)39-s + (−0.987 + 0.156i)41-s + (0.923 + 0.382i)43-s + ⋯
L(s)  = 1  + (0.972 + 0.233i)3-s + (0.156 − 0.987i)7-s + (0.891 + 0.453i)9-s + (0.760 − 0.649i)13-s + (0.309 − 0.951i)17-s + (0.972 + 0.233i)19-s + (0.382 − 0.923i)21-s + (0.707 + 0.707i)23-s + (0.760 + 0.649i)27-s + (−0.852 − 0.522i)29-s + (0.309 + 0.951i)31-s + (0.233 + 0.972i)37-s + (0.891 − 0.453i)39-s + (−0.987 + 0.156i)41-s + (0.923 + 0.382i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.263i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.263i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3520\)    =    \(2^{6} \cdot 5 \cdot 11\)
Sign: $0.964 - 0.263i$
Analytic conductor: \(16.3468\)
Root analytic conductor: \(16.3468\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3520} (1467, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3520,\ (0:\ ),\ 0.964 - 0.263i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.050227951 - 0.4095716450i\)
\(L(\frac12)\) \(\approx\) \(3.050227951 - 0.4095716450i\)
\(L(1)\) \(\approx\) \(1.702414681 - 0.08649150561i\)
\(L(1)\) \(\approx\) \(1.702414681 - 0.08649150561i\)

Euler product

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

−18.71545727590900454991101231780, −18.44242661043224510324266226901, −17.50915341717155386676900855100, −16.63154068815181959178352367990, −15.78073047699301225697824798403, −15.328768266087176606084772185497, −14.52400681454404140808706989245, −14.12496243635058225983790918253, −13.11015143631500184429290434463, −12.75358666735615545306652211503, −11.85386925873937209094291938256, −11.199974982710949214145536904249, −10.24884198466344691029181327745, −9.38698683927195428657272321007, −8.86596899449834538168266063988, −8.33538321163004156800835311044, −7.50975472341846441452939574103, −6.75013405801730524155475123418, −5.92567247003471567150307859292, −5.15728222325745134480769291201, −4.06499853563468570591538273374, −3.49319002914458252330323135167, −2.5409616317103677642662858552, −1.92461618180288725870003342402, −1.041271637613735439070581231725, 0.97010029496731990084391575133, 1.54601048823241882774914717352, 2.9240482407157685996186422861, 3.26407922998812284063088056130, 4.13423422998311385394624205269, 4.89698731015263121587202629670, 5.71922046662032580738669630062, 6.896756960142547215769059285263, 7.49465995130935162860238512616, 8.00370607402083745792076583003, 8.872269507709747489771334466790, 9.62137028321690905696025252842, 10.16269075901742392064656463931, 10.95224332763231460452129322037, 11.6239760173629840544329949584, 12.72026960821499208945437811448, 13.36461679589593599579670788077, 13.946672513989953814926068588219, 14.33570736498964241726130824228, 15.48273555996636977138448307486, 15.68093953156533155761172503812, 16.62276850046589460192903516828, 17.25627473647306949901003096737, 18.2196291331899675340560669823, 18.67524235055617205864965995994

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