Properties

Label 1-4560-4560.59-r1-0-0
Degree $1$
Conductor $4560$
Sign $0.0667 + 0.997i$
Analytic cond. $490.040$
Root an. cond. $490.040$
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 + (−0.984 − 0.173i)13-s + (−0.939 − 0.342i)17-s + (0.766 + 0.642i)23-s + (0.342 + 0.939i)29-s + (−0.5 + 0.866i)31-s + i·37-s + (−0.173 − 0.984i)41-s + (0.642 + 0.766i)43-s + (0.939 − 0.342i)47-s + (−0.5 − 0.866i)49-s + (0.642 − 0.766i)53-s + (−0.342 + 0.939i)59-s + (0.642 − 0.766i)61-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)7-s + (0.866 − 0.5i)11-s + (−0.984 − 0.173i)13-s + (−0.939 − 0.342i)17-s + (0.766 + 0.642i)23-s + (0.342 + 0.939i)29-s + (−0.5 + 0.866i)31-s + i·37-s + (−0.173 − 0.984i)41-s + (0.642 + 0.766i)43-s + (0.939 − 0.342i)47-s + (−0.5 − 0.866i)49-s + (0.642 − 0.766i)53-s + (−0.342 + 0.939i)59-s + (0.642 − 0.766i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0667 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0667 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4560\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.0667 + 0.997i$
Analytic conductor: \(490.040\)
Root analytic conductor: \(490.040\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4560} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4560,\ (1:\ ),\ 0.0667 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8907745726 + 0.8331727776i\)
\(L(\frac12)\) \(\approx\) \(0.8907745726 + 0.8331727776i\)
\(L(1)\) \(\approx\) \(1.033512866 - 0.07748038667i\)
\(L(1)\) \(\approx\) \(1.033512866 - 0.07748038667i\)

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

−17.72350583596957879395220966329, −17.34597363635816430424803233008, −16.700689101593718801227852330376, −15.7610444390611379987125861279, −15.05041697623442306021445582616, −14.73121970259052668831294876946, −13.99706814561192708183815633353, −13.05654807261770342292089488594, −12.40891861937943730708893677750, −11.84474483869933604275209129150, −11.21146242754414714135835502633, −10.45036242403134583694394419747, −9.45219368829076981403393031221, −9.11751473273062461571104570207, −8.341282362789418016240750965740, −7.49895934933479696591962230699, −6.81887833337584268874907795001, −6.05609944130786316162810545293, −5.29868631579294566492592924573, −4.44652033792722030549570799393, −4.01804615980297626314372764979, −2.59601894620824731854719050940, −2.293539218169637900649540957294, −1.32700635765032562158363193879, −0.1946352723243309109593842927, 0.83797405361439387667084323454, 1.50944106345643861195030223308, 2.53335776029986447477750861683, 3.411193140292724911714831504026, 4.14488045058156425899340936684, 4.91442914346603576990493062988, 5.49544502153155576759170198403, 6.83668694038013793905815319226, 6.89632043227849560086425615999, 7.84310163206466385199587039824, 8.679970480195067775843112705078, 9.2622976881421509391973203509, 10.07648839489617728058645044363, 10.864313815829200782455746898793, 11.30424094667595463014018390472, 12.12036698590100299664143176440, 12.801416927457434063866052609501, 13.74465852275378196388673019383, 14.04543648377433894977700376278, 14.84089078540185837311637753782, 15.44429846139471625842308644024, 16.39917198128787667487510943749, 16.91127649319413725472120638206, 17.51511631695906974088507517388, 18.03447899602978586405086021433

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