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Properties

Label 1-4560-4560.2549-r0-0-0
Degree $1$
Conductor $4560$
Sign $-0.843 + 0.536i$
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  + (−0.5 + 0.866i)7-s + (−0.866 + 0.5i)11-s + (0.642 − 0.766i)13-s + (0.173 + 0.984i)17-s + (0.939 − 0.342i)23-s + (−0.984 − 0.173i)29-s + (0.5 − 0.866i)31-s + i·37-s + (−0.766 + 0.642i)41-s + (−0.342 + 0.939i)43-s + (0.173 − 0.984i)47-s + (−0.5 − 0.866i)49-s + (0.342 + 0.939i)53-s + (−0.984 + 0.173i)59-s + (0.342 + 0.939i)61-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)7-s + (−0.866 + 0.5i)11-s + (0.642 − 0.766i)13-s + (0.173 + 0.984i)17-s + (0.939 − 0.342i)23-s + (−0.984 − 0.173i)29-s + (0.5 − 0.866i)31-s + i·37-s + (−0.766 + 0.642i)41-s + (−0.342 + 0.939i)43-s + (0.173 − 0.984i)47-s + (−0.5 − 0.866i)49-s + (0.342 + 0.939i)53-s + (−0.984 + 0.173i)59-s + (0.342 + 0.939i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.843 + 0.536i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.843 + 0.536i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4560\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.843 + 0.536i$
Analytic conductor: \(21.1765\)
Root analytic conductor: \(21.1765\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4560} (2549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4560,\ (0:\ ),\ -0.843 + 0.536i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2231363703 + 0.7664772216i\)
\(L(\frac12)\) \(\approx\) \(0.2231363703 + 0.7664772216i\)
\(L(1)\) \(\approx\) \(0.8637810082 + 0.1947096833i\)
\(L(1)\) \(\approx\) \(0.8637810082 + 0.1947096833i\)

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 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 + (0.642 - 0.766i)T \)
17 \( 1 + (0.173 + 0.984i)T \)
23 \( 1 + (0.939 - 0.342i)T \)
29 \( 1 + (-0.984 - 0.173i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + iT \)
41 \( 1 + (-0.766 + 0.642i)T \)
43 \( 1 + (-0.342 + 0.939i)T \)
47 \( 1 + (0.173 - 0.984i)T \)
53 \( 1 + (0.342 + 0.939i)T \)
59 \( 1 + (-0.984 + 0.173i)T \)
61 \( 1 + (0.342 + 0.939i)T \)
67 \( 1 + (-0.984 - 0.173i)T \)
71 \( 1 + (0.939 + 0.342i)T \)
73 \( 1 + (0.766 - 0.642i)T \)
79 \( 1 + (-0.766 + 0.642i)T \)
83 \( 1 + (0.866 + 0.5i)T \)
89 \( 1 + (-0.766 - 0.642i)T \)
97 \( 1 + (0.173 + 0.984i)T \)
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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

−18.04857807180008665901390409142, −17.12694194706731395686902261062, −16.621911063290853015924687292, −15.91784061806387484533357694185, −15.53876378944063896921335430307, −14.380764953382859579007667764099, −13.85703249859216013151432404047, −13.32107538614509627141741277080, −12.69579090374693590079735922615, −11.78063955648585774427265498401, −11.00171568071661677107003565793, −10.623033104204753802351185041175, −9.70820594634485438844860990641, −9.10318386645538386098580313134, −8.356139353661505961171968156324, −7.39108830529629348927103110141, −7.02504425018087319926410189085, −6.16483527873391645386878117632, −5.311994172733475855053598626637, −4.66160864745957730313108443956, −3.61558394232359500199856634872, −3.232430325255087261569630979039, −2.18525180829579404393179030341, −1.18028596041902322996273640001, −0.2358500429934696185814769443, 1.13021775152883065708955876790, 2.12630430315141512171795178475, 2.88072828996453390454662147917, 3.5126310743633858944703212988, 4.54388914285765023931969097522, 5.33305821561537748785554604196, 5.9580762101037471434879930163, 6.58576288846080424889091301040, 7.592965632456567519893904103467, 8.22547108708131743335983558886, 8.83138601812307754960734167787, 9.713083333873805382450125958546, 10.278608475277696427035227167426, 10.98577376150007928645121372900, 11.78606852220573610884868301578, 12.54943483624253091061414219452, 13.116771364816994606542663661657, 13.48492501473210288981018315994, 14.80130057632106125664900930515, 15.25220600360334993168014510611, 15.53266993539773714843034966362, 16.62051939046756230401780545771, 17.01652523918889533540029013797, 18.06738301254905352015099150333, 18.451694863442197087259766463569

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