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Properties

Label 1-2475-2475.2459-r0-0-0
Degree $1$
Conductor $2475$
Sign $0.342 - 0.939i$
Analytic cond. $11.4938$
Root an. cond. $11.4938$
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  + (0.104 − 0.994i)2-s + (−0.978 − 0.207i)4-s + (−0.978 + 0.207i)7-s + (−0.309 + 0.951i)8-s + (−0.978 − 0.207i)13-s + (0.104 + 0.994i)14-s + (0.913 + 0.406i)16-s + (−0.309 − 0.951i)17-s + (−0.309 + 0.951i)19-s + (−0.978 − 0.207i)23-s + (−0.309 + 0.951i)26-s + 28-s + (0.669 − 0.743i)29-s + (−0.5 + 0.866i)31-s + (0.5 − 0.866i)32-s + ⋯
L(s)  = 1  + (0.104 − 0.994i)2-s + (−0.978 − 0.207i)4-s + (−0.978 + 0.207i)7-s + (−0.309 + 0.951i)8-s + (−0.978 − 0.207i)13-s + (0.104 + 0.994i)14-s + (0.913 + 0.406i)16-s + (−0.309 − 0.951i)17-s + (−0.309 + 0.951i)19-s + (−0.978 − 0.207i)23-s + (−0.309 + 0.951i)26-s + 28-s + (0.669 − 0.743i)29-s + (−0.5 + 0.866i)31-s + (0.5 − 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2475\)    =    \(3^{2} \cdot 5^{2} \cdot 11\)
Sign: $0.342 - 0.939i$
Analytic conductor: \(11.4938\)
Root analytic conductor: \(11.4938\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2475} (2459, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2475,\ (0:\ ),\ 0.342 - 0.939i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6740980888 - 0.4714854824i\)
\(L(\frac12)\) \(\approx\) \(0.6740980888 - 0.4714854824i\)
\(L(1)\) \(\approx\) \(0.6786120595 - 0.3429569418i\)
\(L(1)\) \(\approx\) \(0.6786120595 - 0.3429569418i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.104 - 0.994i)T \)
7 \( 1 + (-0.978 + 0.207i)T \)
13 \( 1 + (-0.978 - 0.207i)T \)
17 \( 1 + (-0.309 - 0.951i)T \)
19 \( 1 + (-0.309 + 0.951i)T \)
23 \( 1 + (-0.978 - 0.207i)T \)
29 \( 1 + (0.669 - 0.743i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (0.809 + 0.587i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (-0.104 + 0.994i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (-0.913 + 0.406i)T \)
61 \( 1 + (0.978 - 0.207i)T \)
67 \( 1 + (-0.913 + 0.406i)T \)
71 \( 1 - T \)
73 \( 1 + T \)
79 \( 1 + (0.978 - 0.207i)T \)
83 \( 1 + (-0.669 + 0.743i)T \)
89 \( 1 + (-0.309 - 0.951i)T \)
97 \( 1 + (0.978 - 0.207i)T \)
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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

−19.681733618752011176976596270812, −18.81483865969306793227175035139, −18.0316220099571689215196636681, −17.29436044967772677080427879591, −16.721951923216456450988045386217, −16.09731211032356809216413604164, −15.31580691687045864945589342631, −14.79262076411697875913211248468, −13.92271789147836824832410479352, −13.24191297790234945561159848497, −12.64798798571700910616177831506, −11.94984137199870297762755465417, −10.718798294732297665585319142611, −9.95448393317174808055070393478, −9.31568140497037884730673962965, −8.60746203619147366883399111046, −7.68353504907789824957142133764, −7.00815932303655699744019233608, −6.35250152099370764846229594327, −5.664941021181570808844761464203, −4.66840608270157981215206380, −4.0116380762233822528949815122, −3.13903393022150285368737783341, −2.06467750044071299366223313713, −0.50817051833794448429507075320, 0.50917911136240234435315476637, 1.81463099798911900190416753327, 2.64117524132305302145568820051, 3.26384334915607058979759282902, 4.20505468083006461407684305862, 4.96377688082274806140622595580, 5.85015183721607657540625952518, 6.64245168430054891948300865137, 7.72206673825019473702803589366, 8.5276041472178900354420685512, 9.41515817297302762572987174556, 9.97469606729996692411081286367, 10.44334363554418711541677165668, 11.62402032409303151855735134481, 12.07188755593477437265819940558, 12.74829598907241845370329837673, 13.430500769208860887912053263566, 14.189726811927609532941200968418, 14.86352557856140743263437331301, 15.80399823446878092282306762862, 16.55928766805209102408863741187, 17.3231405438337925834967573047, 18.23347607170988853194393249386, 18.63933814040555367591631776698, 19.63752229096568437872388835983

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