Properties

Label 2-1875-5.4-c1-0-13
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  + 0.618i·2-s + i·3-s + 1.61·4-s − 0.618·6-s + 2i·7-s + 2.23i·8-s − 9-s − 3·11-s + 1.61i·12-s i·13-s − 1.23·14-s + 1.85·16-s − 0.236i·17-s − 0.618i·18-s − 6.70·19-s + ⋯
L(s)  = 1  + 0.437i·2-s + 0.577i·3-s + 0.809·4-s − 0.252·6-s + 0.755i·7-s + 0.790i·8-s − 0.333·9-s − 0.904·11-s + 0.467i·12-s − 0.277i·13-s − 0.330·14-s + 0.463·16-s − 0.0572i·17-s − 0.145i·18-s − 1.53·19-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

\(L(1)\) \(\approx\) \(1.312411005\)
\(L(\frac12)\) \(\approx\) \(1.312411005\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.618iT - 2T^{2} \)
7 \( 1 - 2iT - 7T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 + iT - 13T^{2} \)
17 \( 1 + 0.236iT - 17T^{2} \)
19 \( 1 + 6.70T + 19T^{2} \)
23 \( 1 - 7.61iT - 23T^{2} \)
29 \( 1 - 1.38T + 29T^{2} \)
31 \( 1 + 4.70T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 + 11.6T + 41T^{2} \)
43 \( 1 + 9.61iT - 43T^{2} \)
47 \( 1 - 9.23iT - 47T^{2} \)
53 \( 1 - 6.76iT - 53T^{2} \)
59 \( 1 - 13.9T + 59T^{2} \)
61 \( 1 + 4.70T + 61T^{2} \)
67 \( 1 + 9.18iT - 67T^{2} \)
71 \( 1 + 1.09T + 71T^{2} \)
73 \( 1 - 2.29iT - 73T^{2} \)
79 \( 1 - 15.8T + 79T^{2} \)
83 \( 1 - 9iT - 83T^{2} \)
89 \( 1 + 11.1T + 89T^{2} \)
97 \( 1 - 2.85iT - 97T^{2} \)
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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

−9.588918047029381318857274067614, −8.686123507184514160921969591919, −8.093174114011152601551294457686, −7.27134843999938928957285534591, −6.37132880279268762045785523158, −5.55092566155764734104371721170, −5.09457328755616601330751222291, −3.74930003170665451640786100248, −2.75666796808375994024581170538, −1.91916599832901717469399012794, 0.41977842533846591014964943112, 1.83194035763247316576318712521, 2.55312829957554503856751277856, 3.63878777804347991647917846809, 4.61971934042264101553154085310, 5.78678932456484747606989159406, 6.75517523272380675154696923870, 7.00607146701682031854145591107, 8.088592876974064850635118439011, 8.606037261840449960165868714753

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