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Properties

Label 1-4560-4560.3197-r0-0-0
Degree $1$
Conductor $4560$
Sign $0.594 - 0.804i$
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  + (0.866 + 0.5i)7-s + (0.866 − 0.5i)11-s + (−0.939 + 0.342i)13-s + (−0.642 − 0.766i)17-s + (−0.984 − 0.173i)23-s + (−0.642 + 0.766i)29-s + (−0.5 + 0.866i)31-s + 37-s + (−0.939 − 0.342i)41-s + (−0.173 − 0.984i)43-s + (0.642 − 0.766i)47-s + (0.5 + 0.866i)49-s + (0.173 − 0.984i)53-s + (−0.642 − 0.766i)59-s + (0.984 + 0.173i)61-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)7-s + (0.866 − 0.5i)11-s + (−0.939 + 0.342i)13-s + (−0.642 − 0.766i)17-s + (−0.984 − 0.173i)23-s + (−0.642 + 0.766i)29-s + (−0.5 + 0.866i)31-s + 37-s + (−0.939 − 0.342i)41-s + (−0.173 − 0.984i)43-s + (0.642 − 0.766i)47-s + (0.5 + 0.866i)49-s + (0.173 − 0.984i)53-s + (−0.642 − 0.766i)59-s + (0.984 + 0.173i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.594 - 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.594 - 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4560\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.594 - 0.804i$
Analytic conductor: \(21.1765\)
Root analytic conductor: \(21.1765\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4560} (3197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4560,\ (0:\ ),\ 0.594 - 0.804i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.412428041 - 0.7126882052i\)
\(L(\frac12)\) \(\approx\) \(1.412428041 - 0.7126882052i\)
\(L(1)\) \(\approx\) \(1.094260998 - 0.06680234966i\)
\(L(1)\) \(\approx\) \(1.094260998 - 0.06680234966i\)

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

−18.132665200879755071274153047, −17.624258595426185758291256851, −17.005371031870172436449765849931, −16.59548947164197785895330366250, −15.40315743228177586752866315846, −14.929000804898540119151802323482, −14.43805507664828719988091389388, −13.63594290056826615134991787265, −12.962201538525411355203084994388, −12.1482314262290632441263208077, −11.559940269962579271628506555100, −10.91786073833409079002734672261, −10.09133225928253258240885828935, −9.529884271729929577771163900038, −8.72169457441370846668133305919, −7.72111400964140506292075885157, −7.58132764573836215255407239835, −6.48035462164595499794152041388, −5.87425524201397717542318322679, −4.8517476328446816462773368450, −4.27705496979561576114968338233, −3.70420834349304261322603068815, −2.41397356209607516293483475481, −1.86104659490566786095647306957, −0.92369901676685927674052365333, 0.47409442800158142471971549696, 1.76081809026083799495933944703, 2.16953468926915632740530801885, 3.25488686200932670874497133618, 4.069148748002776275943519744454, 4.90513108308468422068486141479, 5.408170244860286293641191825282, 6.3749569138774513707996128366, 7.059604194686983708451743251617, 7.76629642813887988007255002253, 8.70378728639360869733141150849, 9.03951087516604473572648773046, 9.88738310290524696526811535990, 10.73856878589244393451054017973, 11.50778332520305491523801142310, 11.90117663034371643527175080327, 12.56522270286687016845505777306, 13.58605039303449664315294704148, 14.19259653817347970063025888156, 14.67775312482063787530728998352, 15.35671143576715977454506009407, 16.21098328725428051758079357310, 16.78687713179552999767372984489, 17.4826385711735941354680340191, 18.1721563968878329826190519867

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