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Properties

Label 1-4560-4560.2477-r0-0-0
Degree $1$
Conductor $4560$
Sign $0.951 - 0.308i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.951 - 0.308i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.951 - 0.308i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4560\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.951 - 0.308i$
Analytic conductor: \(21.1765\)
Root analytic conductor: \(21.1765\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4560} (2477, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4560,\ (0:\ ),\ 0.951 - 0.308i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.678350985 - 0.2654961513i\)
\(L(\frac12)\) \(\approx\) \(1.678350985 - 0.2654961513i\)
\(L(1)\) \(\approx\) \(1.084136940 - 0.08474514389i\)
\(L(1)\) \(\approx\) \(1.084136940 - 0.08474514389i\)

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

−18.36315898235871547958081803927, −17.68603365829427182409644990396, −16.77493463008217797748274416476, −16.20257615498921999657478564567, −15.76534702788636023306660163393, −14.76881443374007578573948411677, −14.2635827838386950931059465849, −13.69921527376322160738085477867, −12.74286680872663524413686339772, −12.02657824672189558492236538933, −11.68144954059117391707413659652, −10.863893317472492990750929621391, −9.90570723773850858771048299132, −9.44908221990941148316755520794, −8.500249446753794140182709073036, −8.19098446385107016030601625571, −7.14855868985119939385999533921, −6.33914055469867493218834775289, −5.78773949071904152798709599531, −4.98819574016450374009051905098, −4.26436806242270374639504050834, −3.19733798893936852231415151710, −2.65285019809844745961048501797, −1.78112835176149351454999036746, −0.70922993561014056708746376792, 0.68420329822737500356414038222, 1.53002047922279043921277805441, 2.5049817265285142553723077781, 3.34959452835024505311828884953, 4.17074493906180619031934252814, 4.76252401250087670880117838772, 5.627984560641239464583119054644, 6.45850823340272317182338743650, 7.20153665268420260943972723406, 7.90357670269761248253313850939, 8.278992502539369025315936790063, 9.67748750543253536639712262540, 9.98766331416899581794631678997, 10.462186570152820116139797472479, 11.51172360536393637212175026043, 12.08941653533946355051691353655, 12.898582540736299988338308157897, 13.351463320931289219988054179589, 14.33351700132067133481228551775, 14.686228906079855849935301765078, 15.5454091947808897461743061231, 16.172155009598836511287107530116, 16.991710230879494933733076438651, 17.604555691484376543285235623121, 17.86581342373592789034116899439

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