Properties

Label 1-4560-4560.1037-r0-0-0
Degree $1$
Conductor $4560$
Sign $-0.593 + 0.804i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.593 + 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.593 + 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4560\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.593 + 0.804i$
Analytic conductor: \(21.1765\)
Root analytic conductor: \(21.1765\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4560} (1037, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4560,\ (0:\ ),\ -0.593 + 0.804i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3547221663 + 0.7024295385i\)
\(L(\frac12)\) \(\approx\) \(0.3547221663 + 0.7024295385i\)
\(L(1)\) \(\approx\) \(0.9045692212 + 0.09013945574i\)
\(L(1)\) \(\approx\) \(0.9045692212 + 0.09013945574i\)

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

−18.040137925554319670741517833084, −17.36460664217178961218777025253, −16.62192635892006422709430732039, −15.93286466540273799052803325210, −15.25696654371891647653908173441, −14.814161620583108227512054857512, −13.92841336177613935924333504113, −13.18513265741991812901843532737, −12.659700290495796423918793127010, −11.844589233546216093988839970881, −11.23015330509729989976209753224, −10.614685709418730864243024102107, −9.688022639519078975058767012815, −9.00842743035496834681298141586, −8.456783818282896075140724844767, −7.74995233092079922161108360942, −6.81286354066269985921906171602, −6.12546207557007495427703066348, −5.37103887256604660956883046770, −4.85226806260107350519009649586, −3.79331740672797790317181923005, −2.77830084238795985322562129171, −2.56240151787721358671105291947, −1.28676127131340472772797469617, −0.22185645480382452305007337034, 1.13174714433615616432964385172, 1.92470170880917740938265878130, 2.738466856327255217961182092459, 3.83272287922242963024466173286, 4.43440061814076956797279314186, 4.90916311965206515125804518319, 6.13790650412397149392358142345, 6.72385101809116686646629453873, 7.47151972160679628098757544683, 7.88805052101325490317110428689, 9.18496415096318091280034740564, 9.41596733632690535405242744494, 10.365322461568285091122189560543, 10.935172803275217703001016328161, 11.64310507148508965327731312945, 12.45113865414289029000229523597, 13.09575578464038443406344838786, 13.76597489071596184591096691242, 14.36793150622658173060338351601, 15.22377694933943186568850850781, 15.59508165699783991835669829015, 16.75459646583180839736499731535, 17.06583905948597396615467938333, 17.602672410424686347744526010194, 18.46702506817736800891767453853

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