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Properties

Label 1-4560-4560.3203-r1-0-0
Degree $1$
Conductor $4560$
Sign $-0.890 + 0.454i$
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  − 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.890 + 0.454i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.890 + 0.454i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4560\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.890 + 0.454i$
Analytic conductor: \(490.040\)
Root analytic conductor: \(490.040\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4560} (3203, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4560,\ (1:\ ),\ -0.890 + 0.454i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2659616555 - 1.106878810i\)
\(L(\frac12)\) \(\approx\) \(-0.2659616555 - 1.106878810i\)
\(L(1)\) \(\approx\) \(0.9507829930 - 0.3664639718i\)
\(L(1)\) \(\approx\) \(0.9507829930 - 0.3664639718i\)

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

−18.59265845600159356573994815187, −17.88623745933057368651711111957, −16.99276037271038714909125736327, −16.52961411011699159464240691312, −15.69147840545379532761575812736, −14.97213452244763327611643432027, −14.659357071406123792211111510815, −13.746359994064843901469088630467, −12.9063250536647223206309417180, −12.32361684991169438278713299392, −11.8352503273427206148870091130, −10.96116315045450961232208537027, −10.301910954172713279545078409088, −9.38491301709298595877016844282, −8.95427849239233419443199643726, −8.25711129149417164919645353424, −7.24136720405996431657372600444, −6.804922072669311719278412027561, −5.74512593562353985201399282119, −5.3320317583634848468168132174, −4.36832770934789574911799823392, −3.64648489384442884666657234265, −2.716793639482810373001181887926, −1.90141775350458907450041894816, −1.28191418743427747111729098300, 0.1834147013785989284912630226, 0.846896420266170023180568580377, 1.62277166827374069282358604007, 2.92121190313654621810606467594, 3.478677401496906296431333058621, 4.083386652897579701080999221540, 5.313684219784120652064640785073, 5.59191192759012667242001225450, 6.63530118805542806990245274890, 7.379871455276887250705376047201, 7.91736770883734144166560700580, 8.718884563506321798350404081826, 9.49127666378867094677439152092, 10.25363901270131147600748811490, 10.947461271271008119451750214477, 11.322511958754214265691190041643, 12.33151188271438274911228513737, 13.12003093412830049608824740810, 13.5694885102580097999068138654, 14.25724950532672004138976709292, 14.904260873588813225447305260, 15.881776763914283708240472208279, 16.23860877406593557816724900488, 17.178152350442301311782077101543, 17.398127590042428123871241800172

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