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Properties

Label 1-2415-2415.398-r0-0-0
Degree $1$
Conductor $2415$
Sign $0.649 - 0.760i$
Analytic cond. $11.2152$
Root an. cond. $11.2152$
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  + (0.989 − 0.142i)2-s + (0.959 − 0.281i)4-s + (0.909 − 0.415i)8-s + (−0.142 + 0.989i)11-s + (0.540 − 0.841i)13-s + (0.841 − 0.540i)16-s + (0.281 − 0.959i)17-s + (0.959 − 0.281i)19-s + i·22-s + (0.415 − 0.909i)26-s + (−0.959 − 0.281i)29-s + (−0.415 − 0.909i)31-s + (0.755 − 0.654i)32-s + (0.142 − 0.989i)34-s + (0.755 − 0.654i)37-s + (0.909 − 0.415i)38-s + ⋯
L(s)  = 1  + (0.989 − 0.142i)2-s + (0.959 − 0.281i)4-s + (0.909 − 0.415i)8-s + (−0.142 + 0.989i)11-s + (0.540 − 0.841i)13-s + (0.841 − 0.540i)16-s + (0.281 − 0.959i)17-s + (0.959 − 0.281i)19-s + i·22-s + (0.415 − 0.909i)26-s + (−0.959 − 0.281i)29-s + (−0.415 − 0.909i)31-s + (0.755 − 0.654i)32-s + (0.142 − 0.989i)34-s + (0.755 − 0.654i)37-s + (0.909 − 0.415i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.649 - 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.649 - 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2415\)    =    \(3 \cdot 5 \cdot 7 \cdot 23\)
Sign: $0.649 - 0.760i$
Analytic conductor: \(11.2152\)
Root analytic conductor: \(11.2152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2415} (398, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2415,\ (0:\ ),\ 0.649 - 0.760i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.214077033 - 1.482169003i\)
\(L(\frac12)\) \(\approx\) \(3.214077033 - 1.482169003i\)
\(L(1)\) \(\approx\) \(2.040359934 - 0.4127188923i\)
\(L(1)\) \(\approx\) \(2.040359934 - 0.4127188923i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.989 - 0.142i)T \)
11 \( 1 + (-0.142 + 0.989i)T \)
13 \( 1 + (0.540 - 0.841i)T \)
17 \( 1 + (0.281 - 0.959i)T \)
19 \( 1 + (0.959 - 0.281i)T \)
29 \( 1 + (-0.959 - 0.281i)T \)
31 \( 1 + (-0.415 - 0.909i)T \)
37 \( 1 + (0.755 - 0.654i)T \)
41 \( 1 + (-0.654 + 0.755i)T \)
43 \( 1 + (-0.909 - 0.415i)T \)
47 \( 1 - iT \)
53 \( 1 + (0.540 + 0.841i)T \)
59 \( 1 + (-0.841 - 0.540i)T \)
61 \( 1 + (0.415 + 0.909i)T \)
67 \( 1 + (0.989 - 0.142i)T \)
71 \( 1 + (0.142 + 0.989i)T \)
73 \( 1 + (0.281 + 0.959i)T \)
79 \( 1 + (0.841 + 0.540i)T \)
83 \( 1 + (-0.755 + 0.654i)T \)
89 \( 1 + (0.415 - 0.909i)T \)
97 \( 1 + (0.755 + 0.654i)T \)
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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

−19.767197666686023273605289998298, −18.97108860561352188199844329284, −18.344067705068028068399727861829, −17.23443747068919484673535537552, −16.44686803417864867584933512791, −16.170188829526182272600205424373, −15.2140162189895253262809630381, −14.51807504842881313035479181543, −13.83831114795605506364702216292, −13.30850831095319662381247964810, −12.48962076824142156379057164825, −11.72226441505536699354672008041, −11.10348739947960138624139987200, −10.45126148023735207224473341876, −9.3737412311502626631681462609, −8.42344568954218688460390695625, −7.77554624465778598432073456268, −6.81592247012031254686436031782, −6.13752960856924733633128458534, −5.470314744426797621706368699757, −4.65797890182854997124211232332, −3.53650310488459449871096850707, −3.35575303729387755843455533013, −2.020183365638521096617121702560, −1.25691165603795147704211733889, 0.843031765875212654636063845002, 1.93551727938777608587744553531, 2.76000416645310612966961855866, 3.55925676616628119260801893873, 4.3773828503559148847394399574, 5.30452441580449653134866607563, 5.69130771331815482996760551137, 6.86404268621705436830316446526, 7.39251048200730014115453527863, 8.16191212787245876077143468394, 9.51376739869531049437586773474, 9.95359804087801454692680440635, 10.95517679585254479133619072196, 11.565014854455364302024481634127, 12.2653862209989020386565142099, 13.09656282696439750224128988746, 13.5068107063914799527383577626, 14.40729129885802468062909317141, 15.1537758385386117403948776447, 15.5918572368322165158147357376, 16.42497268184507848966921728426, 17.1347377479993223201072793865, 18.2552811406062676800100677719, 18.55762382934884345607800089756, 19.89879082068760917669946884687

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