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Properties

Label 1-2475-2475.2362-r0-0-0
Degree $1$
Conductor $2475$
Sign $-0.733 - 0.679i$
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  + (0.866 + 0.5i)2-s + (0.5 + 0.866i)4-s + (−0.406 − 0.913i)7-s + i·8-s + (−0.994 − 0.104i)13-s + (0.104 − 0.994i)14-s + (−0.5 + 0.866i)16-s + (−0.951 − 0.309i)17-s + 19-s + (−0.743 − 0.669i)23-s + (−0.809 − 0.587i)26-s + (0.587 − 0.809i)28-s + (−0.5 + 0.866i)29-s + (0.913 + 0.406i)31-s + (−0.866 + 0.5i)32-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.5 + 0.866i)4-s + (−0.406 − 0.913i)7-s + i·8-s + (−0.994 − 0.104i)13-s + (0.104 − 0.994i)14-s + (−0.5 + 0.866i)16-s + (−0.951 − 0.309i)17-s + 19-s + (−0.743 − 0.669i)23-s + (−0.809 − 0.587i)26-s + (0.587 − 0.809i)28-s + (−0.5 + 0.866i)29-s + (0.913 + 0.406i)31-s + (−0.866 + 0.5i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.733 - 0.679i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.733 - 0.679i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2475\)    =    \(3^{2} \cdot 5^{2} \cdot 11\)
Sign: $-0.733 - 0.679i$
Analytic conductor: \(11.4938\)
Root analytic conductor: \(11.4938\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2475} (2362, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2475,\ (0:\ ),\ -0.733 - 0.679i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01954863766 - 0.04990278062i\)
\(L(\frac12)\) \(\approx\) \(0.01954863766 - 0.04990278062i\)
\(L(1)\) \(\approx\) \(1.167258454 + 0.3112866258i\)
\(L(1)\) \(\approx\) \(1.167258454 + 0.3112866258i\)

Euler product

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

−19.850757214811495287872706090191, −19.18511992360082764803870881461, −18.55490422053201546333235067634, −17.72960055266626778849382925484, −16.805857139852746764091847394708, −15.86335512256063458182202737969, −15.32632526108465648125719675706, −14.84919158571399501720120487730, −13.76219324902216426452431094731, −13.39163059937266120862895574246, −12.46332252705097289963908702525, −11.792313425890722642818787691833, −11.49937116853603354924619249572, −10.22702441780087895170744219639, −9.78037940896691196228335899248, −9.013452629985441979958825712924, −7.94808848366263438593084924114, −6.96363879929970745649756751280, −6.26008273616420559562139960469, −5.4803264707145774890254869168, −4.83887401104489419272749025844, −3.92012922326219885733472053954, −3.037944376175017952044932278292, −2.332515587800704109701918012417, −1.57137528632761134471181640923, 0.010974084804450875887937732694, 1.61480125911333462127541011436, 2.72248633042896658134038968979, 3.37562851820575233966382032673, 4.34418452892898995490700868145, 4.87439610385507740307144019469, 5.78031416581344206062892014618, 6.77971097417931062873240185307, 7.122490568948489834054036373341, 7.93411392104091124873490633172, 8.82650255046824100537984250034, 9.83001565218020106092872018046, 10.54324097377297235156394796179, 11.44798790315757231307861454505, 12.14638512442744069377612537619, 12.86292541728904318086254223001, 13.59650033755289803975900735096, 14.10860324272286699191047879270, 14.832787934570684585548674677355, 15.657120763113861328031122809258, 16.31337515721861642760193036146, 16.84199785296700742777883968025, 17.65183800873653291106190408727, 18.24263542495040103153906583164, 19.58028783368987807722377123923

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