Properties

Label 1-4560-4560.107-r0-0-0
Degree $1$
Conductor $4560$
Sign $0.998 + 0.0629i$
Analytic cond. $21.1765$
Root an. cond. $21.1765$
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 i·11-s + (−0.5 − 0.866i)13-s + (0.866 + 0.5i)17-s + (0.866 − 0.5i)23-s + (0.866 − 0.5i)29-s + 31-s + 37-s + (0.5 − 0.866i)41-s + (0.5 − 0.866i)43-s + (0.866 − 0.5i)47-s − 49-s + (−0.5 − 0.866i)53-s + (−0.866 − 0.5i)59-s + (−0.866 + 0.5i)61-s + ⋯
L(s)  = 1  + i·7-s i·11-s + (−0.5 − 0.866i)13-s + (0.866 + 0.5i)17-s + (0.866 − 0.5i)23-s + (0.866 − 0.5i)29-s + 31-s + 37-s + (0.5 − 0.866i)41-s + (0.5 − 0.866i)43-s + (0.866 − 0.5i)47-s − 49-s + (−0.5 − 0.866i)53-s + (−0.866 − 0.5i)59-s + (−0.866 + 0.5i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0629i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0629i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4560\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.998 + 0.0629i$
Analytic conductor: \(21.1765\)
Root analytic conductor: \(21.1765\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4560} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4560,\ (0:\ ),\ 0.998 + 0.0629i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.862220601 + 0.05866348185i\)
\(L(\frac12)\) \(\approx\) \(1.862220601 + 0.05866348185i\)
\(L(1)\) \(\approx\) \(1.141885453 + 0.09297534009i\)
\(L(1)\) \(\approx\) \(1.141885453 + 0.09297534009i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
19 \( 1 \)
good7 \( 1 + iT \)
11 \( 1 - iT \)
13 \( 1 + (-0.5 - 0.866i)T \)
17 \( 1 + (0.866 + 0.5i)T \)
23 \( 1 + (0.866 - 0.5i)T \)
29 \( 1 + (0.866 - 0.5i)T \)
31 \( 1 + T \)
37 \( 1 + T \)
41 \( 1 + (0.5 - 0.866i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (0.866 - 0.5i)T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + (-0.866 - 0.5i)T \)
61 \( 1 + (-0.866 + 0.5i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + (-0.866 - 0.5i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 - T \)
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

−18.244327194277576292283246089916, −17.43596033259564259890881445739, −16.735955096819778359079989440530, −16.43672008036755361164654466150, −15.65274815347996624615298049465, −14.70476647246975737276086098178, −14.0161153735780445255292415349, −13.75660976075128858795079195589, −12.85834283036578673026559962159, −12.10166504653562208581387722916, −11.307477596859582026295028138787, −10.87517830703348254823214323429, −9.96478563013947593593384227525, −9.42875135702495466596082721230, −8.61812401560635679550093687261, −7.72904554919162769886027241597, −7.28291243459093359804648582567, −6.406678002954345340814014896946, −5.77688181669642168759496709434, −4.66594484915310404945807552293, −4.33368617750231884203996682709, −3.15048155072699127790188904022, −2.810811764082341965810314610395, −1.3461441056579436903152479763, −0.87292066293965371988073685925, 0.66952996976114763802309166535, 1.78044285249449875465781644295, 2.59627968246964616520340606053, 3.13129387543713544034733401779, 4.30676635773954017458480780605, 4.93307995274664551342037118665, 5.697149452366561698715995119326, 6.31248083702656438385702679778, 7.284668572960575163883026782, 7.88259368270459610027048322994, 8.635026294070718284865205243741, 9.34910147672630754342625233844, 10.102878800084575114751606731152, 10.57215150238251939971955211468, 11.63113924576402434376827106287, 12.34518106201916128711356662278, 12.547328366083423241517448080387, 13.44143645740433692362744176903, 14.40843782972686563705198492410, 14.96029138525061618592239347777, 15.4307472709292657268210501148, 16.08843951787043342772120792856, 17.171888261172293518067717951203, 17.421236275763044474376827257130, 18.26720843561574592808950019017

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