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Properties

Label 1-2520-2520.67-r0-0-0
Degree $1$
Conductor $2520$
Sign $0.927 - 0.373i$
Analytic cond. $11.7028$
Root an. cond. $11.7028$
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  + 11-s + (0.866 − 0.5i)13-s + (0.866 − 0.5i)17-s + (0.5 − 0.866i)19-s − i·23-s + (−0.5 + 0.866i)29-s + (0.5 − 0.866i)31-s + (0.866 + 0.5i)37-s + (−0.5 − 0.866i)41-s + (0.866 + 0.5i)43-s + (−0.866 + 0.5i)47-s + (0.866 − 0.5i)53-s + (0.5 − 0.866i)59-s + (0.5 + 0.866i)61-s + (−0.866 − 0.5i)67-s + ⋯
L(s)  = 1  + 11-s + (0.866 − 0.5i)13-s + (0.866 − 0.5i)17-s + (0.5 − 0.866i)19-s − i·23-s + (−0.5 + 0.866i)29-s + (0.5 − 0.866i)31-s + (0.866 + 0.5i)37-s + (−0.5 − 0.866i)41-s + (0.866 + 0.5i)43-s + (−0.866 + 0.5i)47-s + (0.866 − 0.5i)53-s + (0.5 − 0.866i)59-s + (0.5 + 0.866i)61-s + (−0.866 − 0.5i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.373i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.373i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2520\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.927 - 0.373i$
Analytic conductor: \(11.7028\)
Root analytic conductor: \(11.7028\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2520} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2520,\ (0:\ ),\ 0.927 - 0.373i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.000935645 - 0.3873780805i\)
\(L(\frac12)\) \(\approx\) \(2.000935645 - 0.3873780805i\)
\(L(1)\) \(\approx\) \(1.263205760 - 0.08669215934i\)
\(L(1)\) \(\approx\) \(1.263205760 - 0.08669215934i\)

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 \( 1 + T \)
13 \( 1 + (0.866 - 0.5i)T \)
17 \( 1 + (0.866 - 0.5i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 - iT \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (0.866 + 0.5i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (0.866 + 0.5i)T \)
47 \( 1 + (-0.866 + 0.5i)T \)
53 \( 1 + (0.866 - 0.5i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (0.5 + 0.866i)T \)
67 \( 1 + (-0.866 - 0.5i)T \)
71 \( 1 - T \)
73 \( 1 + (-0.866 + 0.5i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (0.866 + 0.5i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (-0.866 - 0.5i)T \)
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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

−19.35306711725143543789085650678, −18.84083498011036979665371202008, −18.12113431611275093348957441916, −17.31783413516428486998225239274, −16.41309943050410238290481778225, −16.28820847489248072757332046062, −15.009256317044234087006671848513, −14.53579329227068307556033934180, −13.82716895382929578283826367712, −13.06708431380929191050504590917, −12.098615327602984467631963582483, −11.73917971132203832284572524622, −10.767540747357413274013212708953, −10.05532224951618404285878719010, −9.27533761260197000620232339486, −8.51537393610683392636396449131, −7.840815447494073645840467262311, −6.84310607202846995170316145981, −6.166552647508798026141672611907, −5.50963249392097148789026575561, −4.28219634033092230166160543008, −3.82345681138421409074532570067, −2.86351436711079368940252888255, −1.69273509464080120968081474699, −1.01690899125904330908951423080, 0.84413953588025589274209275606, 1.5667703757217620759497815736, 2.85471801007817746018043072029, 3.50975849644693042355157130030, 4.35692282042131608760405874001, 5.36848510890398836202046033303, 5.9812827192540590418985365524, 6.92011530372224162173708660783, 7.59331943155528989482492285212, 8.46355983774652480233034937470, 9.28589173203118547448653981983, 9.79846910316834076771391223376, 10.82201128058898514695875137254, 11.53212286852826388040854058622, 12.021486789882479540412619598399, 13.13866957904014391961003399240, 13.5430104988855462905452523670, 14.47459864204858343748609877750, 15.040206018012137371640775255370, 15.96552579714398063490822183854, 16.43786080628280581950715889465, 17.40638267208520860453167539173, 17.86219024870160983356905160740, 18.74514933951145916280514684393, 19.36243443903906456641820339707

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