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Properties

Label 1-2580-2580.1943-r0-0-0
Degree $1$
Conductor $2580$
Sign $0.362 - 0.931i$
Analytic cond. $11.9814$
Root an. cond. $11.9814$
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  − i·7-s + (0.623 − 0.781i)11-s + (0.974 − 0.222i)13-s + (−0.974 − 0.222i)17-s + (−0.623 − 0.781i)19-s + (0.781 + 0.623i)23-s + (0.900 + 0.433i)29-s + (0.900 + 0.433i)31-s + i·37-s + (0.900 + 0.433i)41-s + (0.781 − 0.623i)47-s − 49-s + (0.974 + 0.222i)53-s + (0.222 − 0.974i)59-s + (0.900 − 0.433i)61-s + ⋯
L(s)  = 1  − i·7-s + (0.623 − 0.781i)11-s + (0.974 − 0.222i)13-s + (−0.974 − 0.222i)17-s + (−0.623 − 0.781i)19-s + (0.781 + 0.623i)23-s + (0.900 + 0.433i)29-s + (0.900 + 0.433i)31-s + i·37-s + (0.900 + 0.433i)41-s + (0.781 − 0.623i)47-s − 49-s + (0.974 + 0.222i)53-s + (0.222 − 0.974i)59-s + (0.900 − 0.433i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2580 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.362 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2580 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.362 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2580\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 43\)
Sign: $0.362 - 0.931i$
Analytic conductor: \(11.9814\)
Root analytic conductor: \(11.9814\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2580} (1943, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2580,\ (0:\ ),\ 0.362 - 0.931i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.494824020 - 1.022238220i\)
\(L(\frac12)\) \(\approx\) \(1.494824020 - 1.022238220i\)
\(L(1)\) \(\approx\) \(1.143771003 - 0.2763335094i\)
\(L(1)\) \(\approx\) \(1.143771003 - 0.2763335094i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
43 \( 1 \)
good7 \( 1 - iT \)
11 \( 1 + (0.623 - 0.781i)T \)
13 \( 1 + (0.974 - 0.222i)T \)
17 \( 1 + (-0.974 - 0.222i)T \)
19 \( 1 + (-0.623 - 0.781i)T \)
23 \( 1 + (0.781 + 0.623i)T \)
29 \( 1 + (0.900 + 0.433i)T \)
31 \( 1 + (0.900 + 0.433i)T \)
37 \( 1 + iT \)
41 \( 1 + (0.900 + 0.433i)T \)
47 \( 1 + (0.781 - 0.623i)T \)
53 \( 1 + (0.974 + 0.222i)T \)
59 \( 1 + (0.222 - 0.974i)T \)
61 \( 1 + (0.900 - 0.433i)T \)
67 \( 1 + (-0.781 + 0.623i)T \)
71 \( 1 + (-0.623 - 0.781i)T \)
73 \( 1 + (-0.974 + 0.222i)T \)
79 \( 1 + T \)
83 \( 1 + (-0.433 - 0.900i)T \)
89 \( 1 + (0.900 - 0.433i)T \)
97 \( 1 + (-0.781 - 0.623i)T \)
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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

−19.24146357868521396412166051945, −19.03522572815145883464416024889, −17.92022630454360902109840203465, −17.650713102211771669214615945985, −16.630871944130398786569226340581, −15.9240111224541495007626775644, −15.198049786038185845932360461668, −14.71274606614458419884823378162, −13.82553619734777514646691572500, −12.986076514003344491289109474705, −12.332190690076453210173676580592, −11.70965132149904681290031949169, −10.886787105015221895852587674, −10.15909788570300805421006823715, −9.09611653122592785581798744865, −8.79807174199741115110671994930, −7.95297807263651369101031662193, −6.84788389087622317865493665875, −6.26866814473076780234885555145, −5.57862353621193331052513924099, −4.413762486388158311011920650576, −4.00999255954994584192257633605, −2.687653169390890665824188155311, −2.104099234785867755877592895, −1.0682892500485965923076790287, 0.6978790972840691832479382178, 1.37235786187060314750634016878, 2.70001321006717605568200616529, 3.482815751587550695471992000889, 4.27652757459215080283356652453, 4.980293451767247411719771492384, 6.178729533779686593259665406402, 6.657723065227216021013389508138, 7.4219867115729736057273641137, 8.55284216862898421047441149994, 8.817300481553830756618799198456, 9.91398736695518556721932257739, 10.75575370430948744381734476122, 11.17408945614275753764754152748, 11.929029854476015968080340652211, 13.17850093198635731465560858132, 13.41667920905487636345080758413, 14.11657534719958309132113441431, 15.00859146842121437151213668, 15.81807453845692577586519680368, 16.355504930868808153079211908590, 17.33407857708821102480545459777, 17.56819358871607160154989370013, 18.59247515805888681027711144077, 19.41351193637132917099289157447

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