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Properties

Label 1-2475-2475.1316-r0-0-0
Degree $1$
Conductor $2475$
Sign $0.361 + 0.932i$
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  + (0.913 + 0.406i)2-s + (0.669 + 0.743i)4-s + (−0.913 − 0.406i)7-s + (0.309 + 0.951i)8-s + (0.5 − 0.866i)13-s + (−0.669 − 0.743i)14-s + (−0.104 + 0.994i)16-s + (−0.809 + 0.587i)17-s + (−0.309 − 0.951i)19-s + (0.978 + 0.207i)23-s + (0.809 − 0.587i)26-s + (−0.309 − 0.951i)28-s + (−0.978 + 0.207i)29-s + (0.669 + 0.743i)31-s + (−0.5 + 0.866i)32-s + ⋯
L(s)  = 1  + (0.913 + 0.406i)2-s + (0.669 + 0.743i)4-s + (−0.913 − 0.406i)7-s + (0.309 + 0.951i)8-s + (0.5 − 0.866i)13-s + (−0.669 − 0.743i)14-s + (−0.104 + 0.994i)16-s + (−0.809 + 0.587i)17-s + (−0.309 − 0.951i)19-s + (0.978 + 0.207i)23-s + (0.809 − 0.587i)26-s + (−0.309 − 0.951i)28-s + (−0.978 + 0.207i)29-s + (0.669 + 0.743i)31-s + (−0.5 + 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.361 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.361 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2475\)    =    \(3^{2} \cdot 5^{2} \cdot 11\)
Sign: $0.361 + 0.932i$
Analytic conductor: \(11.4938\)
Root analytic conductor: \(11.4938\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2475} (1316, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2475,\ (0:\ ),\ 0.361 + 0.932i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.149722215 + 1.472387036i\)
\(L(\frac12)\) \(\approx\) \(2.149722215 + 1.472387036i\)
\(L(1)\) \(\approx\) \(1.590411645 + 0.5123856908i\)
\(L(1)\) \(\approx\) \(1.590411645 + 0.5123856908i\)

Euler product

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

−19.28023908434163077397851165791, −18.88780678860469787435994000775, −18.23843122852504836445795976943, −16.9138218278436908862858077919, −16.35859433276017583714418396937, −15.65272197911568497381959469314, −15.0130164623210683911265335925, −14.2501428468879759732311330260, −13.408018741769918390362878407599, −12.99918322582837411836026240761, −12.19266179191041985875354830800, −11.47052821344419259685299059740, −10.888662341379094901534376762535, −9.88299138164774713937213985243, −9.34934692641400087912678389232, −8.471760645280526408239549159068, −7.17043516398855362645986801686, −6.62628053131823074432485168589, −5.89736447114046544847765941662, −5.16465542668815823475056582890, −4.088467893680781990539621574136, −3.65216619766007190934324159783, −2.53839691557237549886598400760, −2.01555115545494587563853782413, −0.69184991857179450913782111012, 0.992735334974282648680655993437, 2.37444363868544054407855425938, 3.10203647954753684507890999993, 3.821619452542261893782913035951, 4.63029331813084643073341520778, 5.502371695501118510825009483376, 6.30566405244019417361530761432, 6.86974214389977775365646573881, 7.614937332484016511276925966353, 8.567874898381347943529278669564, 9.25137599256146957158270695794, 10.503508528890742340270420782710, 10.88956918509735401631377319812, 11.81616190790021926497991923422, 12.843013788607737051279840137495, 13.09162321318546200866386917588, 13.694593467678422588007738086819, 14.68396462848250598298007678905, 15.46928676302247208184007294331, 15.73443304696762345100493819834, 16.72073229000772614352976413183, 17.2754434571893078982314017595, 17.95684275468368159243295741913, 19.14010210233018031726171557259, 19.72438839763382331008405041630

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