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Properties

Label 1-3360-3360.1859-r0-0-0
Degree $1$
Conductor $3360$
Sign $-0.501 + 0.864i$
Analytic cond. $15.6037$
Root an. cond. $15.6037$
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  + (−0.965 − 0.258i)11-s + (0.707 + 0.707i)13-s + (0.5 + 0.866i)17-s + (−0.965 + 0.258i)19-s + (0.866 + 0.5i)23-s + (−0.707 − 0.707i)29-s + (0.5 + 0.866i)31-s + (0.258 + 0.965i)37-s − i·41-s + (0.707 − 0.707i)43-s + (−0.5 + 0.866i)47-s + (−0.965 − 0.258i)53-s + (0.965 + 0.258i)59-s + (−0.965 + 0.258i)61-s + (0.258 − 0.965i)67-s + ⋯
L(s)  = 1  + (−0.965 − 0.258i)11-s + (0.707 + 0.707i)13-s + (0.5 + 0.866i)17-s + (−0.965 + 0.258i)19-s + (0.866 + 0.5i)23-s + (−0.707 − 0.707i)29-s + (0.5 + 0.866i)31-s + (0.258 + 0.965i)37-s − i·41-s + (0.707 − 0.707i)43-s + (−0.5 + 0.866i)47-s + (−0.965 − 0.258i)53-s + (0.965 + 0.258i)59-s + (−0.965 + 0.258i)61-s + (0.258 − 0.965i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.501 + 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.501 + 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3360\)    =    \(2^{5} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.501 + 0.864i$
Analytic conductor: \(15.6037\)
Root analytic conductor: \(15.6037\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3360} (1859, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3360,\ (0:\ ),\ -0.501 + 0.864i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4841392997 + 0.8405663167i\)
\(L(\frac12)\) \(\approx\) \(0.4841392997 + 0.8405663167i\)
\(L(1)\) \(\approx\) \(0.9224655001 + 0.1576540168i\)
\(L(1)\) \(\approx\) \(0.9224655001 + 0.1576540168i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good11 \( 1 + (-0.965 - 0.258i)T \)
13 \( 1 + (0.707 + 0.707i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-0.965 + 0.258i)T \)
23 \( 1 + (0.866 + 0.5i)T \)
29 \( 1 + (-0.707 - 0.707i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (0.258 + 0.965i)T \)
41 \( 1 - iT \)
43 \( 1 + (0.707 - 0.707i)T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (-0.965 - 0.258i)T \)
59 \( 1 + (0.965 + 0.258i)T \)
61 \( 1 + (-0.965 + 0.258i)T \)
67 \( 1 + (0.258 - 0.965i)T \)
71 \( 1 - iT \)
73 \( 1 + (-0.866 + 0.5i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (0.707 + 0.707i)T \)
89 \( 1 + (-0.866 - 0.5i)T \)
97 \( 1 - T \)
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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

−18.60382143146887493013292203186, −17.95781244023533542225791718544, −17.31380870597700514756271486129, −16.3755387623942321319934935513, −15.95746206516420210578387590997, −15.03382872840441360244132005204, −14.65348559716232315563685586002, −13.576214110263137836337683874288, −12.97879040110511058273089768594, −12.57132496912385879592786440525, −11.42493792076188080503031130574, −10.91967988995108482624226739752, −10.21432709095122903867261628890, −9.443575922839339731951158444109, −8.631998389284150988131757851520, −7.91215945641060841410951762833, −7.274583695180105132873092435146, −6.37553325488463207674559184261, −5.57830263274007922485568278152, −4.90739892444489523374568091174, −4.08124584293228757916079551269, −3.02803119030922756337202819288, −2.524341907590611324471527962006, −1.36303062907093865794776722332, −0.2972249174066857848615246004, 1.171462622162509078479890272443, 1.99048954791292899729066431079, 2.96007468571032965789236448282, 3.75731525803606960867809036192, 4.53542468103074061849206314536, 5.45509517551950315174780022486, 6.11731013803349878736052488155, 6.86535225971457751154309879406, 7.81333424240166380413880246874, 8.38717837633608665102052898447, 9.1034074762259480322763787845, 9.988674504969392454532409928269, 10.74417962504505637485264770064, 11.18455200760984832067972745440, 12.1759894035056150326489636424, 12.833418838392904751510612946201, 13.48971953719742879801124669166, 14.13909787303201399529036663778, 15.04072663424493053974797265542, 15.54827513022971129295906756442, 16.32486425331673506665930769765, 17.0156650400181409680692270420, 17.59078382151292889539803194232, 18.56836225938859665165517413018, 19.01631118937154816292038848671

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