Properties

Label 2-1875-1.1-c1-0-22
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 yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.770·2-s + 3-s − 1.40·4-s − 0.770·6-s + 3.98·7-s + 2.62·8-s + 9-s − 6.35·11-s − 1.40·12-s + 5.35·13-s − 3.07·14-s + 0.793·16-s − 3.45·17-s − 0.770·18-s + 2.32·19-s + 3.98·21-s + 4.89·22-s + 0.955·23-s + 2.62·24-s − 4.12·26-s + 27-s − 5.61·28-s + 7.26·29-s + 6.40·31-s − 5.85·32-s − 6.35·33-s + 2.66·34-s + ⋯
L(s)  = 1  − 0.544·2-s + 0.577·3-s − 0.703·4-s − 0.314·6-s + 1.50·7-s + 0.927·8-s + 0.333·9-s − 1.91·11-s − 0.406·12-s + 1.48·13-s − 0.820·14-s + 0.198·16-s − 0.838·17-s − 0.181·18-s + 0.532·19-s + 0.870·21-s + 1.04·22-s + 0.199·23-s + 0.535·24-s − 0.808·26-s + 0.192·27-s − 1.06·28-s + 1.34·29-s + 1.15·31-s − 1.03·32-s − 1.10·33-s + 0.456·34-s + ⋯

Functional equation

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

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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1875,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.578388290\)
\(L(\frac12)\) \(\approx\) \(1.578388290\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
5 \( 1 \)
good2 \( 1 + 0.770T + 2T^{2} \)
7 \( 1 - 3.98T + 7T^{2} \)
11 \( 1 + 6.35T + 11T^{2} \)
13 \( 1 - 5.35T + 13T^{2} \)
17 \( 1 + 3.45T + 17T^{2} \)
19 \( 1 - 2.32T + 19T^{2} \)
23 \( 1 - 0.955T + 23T^{2} \)
29 \( 1 - 7.26T + 29T^{2} \)
31 \( 1 - 6.40T + 31T^{2} \)
37 \( 1 + 2.83T + 37T^{2} \)
41 \( 1 + 5.35T + 41T^{2} \)
43 \( 1 - 3.93T + 43T^{2} \)
47 \( 1 - 2.48T + 47T^{2} \)
53 \( 1 - 3.95T + 53T^{2} \)
59 \( 1 - 0.0941T + 59T^{2} \)
61 \( 1 + 6.61T + 61T^{2} \)
67 \( 1 + 10.2T + 67T^{2} \)
71 \( 1 + 5.81T + 71T^{2} \)
73 \( 1 - 3.79T + 73T^{2} \)
79 \( 1 + 5.44T + 79T^{2} \)
83 \( 1 - 9.24T + 83T^{2} \)
89 \( 1 - 10.1T + 89T^{2} \)
97 \( 1 - 7.89T + 97T^{2} \)
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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

−8.797484528100568429970181427243, −8.550900567513695091894034688602, −7.940743068118509057651718148692, −7.32763295541994004436168192651, −5.93169724519740317179316018846, −4.87885539136114565002967159812, −4.54493202434009524628866520921, −3.24661801500335696125460887380, −2.07394473359768718010818376466, −0.955196890172590259597388944449, 0.955196890172590259597388944449, 2.07394473359768718010818376466, 3.24661801500335696125460887380, 4.54493202434009524628866520921, 4.87885539136114565002967159812, 5.93169724519740317179316018846, 7.32763295541994004436168192651, 7.940743068118509057651718148692, 8.550900567513695091894034688602, 8.797484528100568429970181427243

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