Properties

Label 2-1875-5.4-c1-0-73
Degree $2$
Conductor $1875$
Sign $-1$
Analytic cond. $14.9719$
Root an. cond. $3.86936$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.858i·2-s i·3-s + 1.26·4-s − 0.858·6-s − 3.88i·7-s − 2.80i·8-s − 9-s − 1.39·11-s − 1.26i·12-s + 3.36i·13-s − 3.33·14-s + 0.118·16-s − 3.11i·17-s + 0.858i·18-s + 2.70·19-s + ⋯
L(s)  = 1  − 0.607i·2-s − 0.577i·3-s + 0.631·4-s − 0.350·6-s − 1.46i·7-s − 0.990i·8-s − 0.333·9-s − 0.421·11-s − 0.364i·12-s + 0.932i·13-s − 0.890·14-s + 0.0296·16-s − 0.755i·17-s + 0.202i·18-s + 0.620·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1875\)    =    \(3 \cdot 5^{4}\)
Sign: $-1$
Analytic conductor: \(14.9719\)
Root analytic conductor: \(3.86936\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1875} (1249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1875,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.698629096\)
\(L(\frac12)\) \(\approx\) \(1.698629096\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
5 \( 1 \)
good2 \( 1 + 0.858iT - 2T^{2} \)
7 \( 1 + 3.88iT - 7T^{2} \)
11 \( 1 + 1.39T + 11T^{2} \)
13 \( 1 - 3.36iT - 13T^{2} \)
17 \( 1 + 3.11iT - 17T^{2} \)
19 \( 1 - 2.70T + 19T^{2} \)
23 \( 1 + 6.43iT - 23T^{2} \)
29 \( 1 + 8.26T + 29T^{2} \)
31 \( 1 + 6.34T + 31T^{2} \)
37 \( 1 - 7.49iT - 37T^{2} \)
41 \( 1 - 11.3T + 41T^{2} \)
43 \( 1 + 3.39iT - 43T^{2} \)
47 \( 1 + 8.38iT - 47T^{2} \)
53 \( 1 - 11.4iT - 53T^{2} \)
59 \( 1 + 7.64T + 59T^{2} \)
61 \( 1 - 10.8T + 61T^{2} \)
67 \( 1 + 3.54iT - 67T^{2} \)
71 \( 1 - 1.18T + 71T^{2} \)
73 \( 1 - 2.39iT - 73T^{2} \)
79 \( 1 + 10.6T + 79T^{2} \)
83 \( 1 - 1.40iT - 83T^{2} \)
89 \( 1 + 4.62T + 89T^{2} \)
97 \( 1 - 1.51iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.966172036799529041679360378344, −7.72310534929202637199000094584, −7.23813631044755459747204526648, −6.75282917086798776845215820114, −5.73715154362307400347722868483, −4.48898038686578982520059296382, −3.67443431221668636156834108118, −2.65795579107958064891750949585, −1.65808630431815450189839825511, −0.57878798407318853756847452099, 1.89452121932186403666510421344, 2.80727633208120810331832811185, 3.71319587261839668624214054660, 5.29666950663478945115464548573, 5.55388269251160249476923961713, 6.15496653843834654610435718456, 7.52980771152222112278022463069, 7.83141948368431556540788438424, 8.884913303740729561788039605671, 9.400003552210869734889323775389

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