Properties

Label 2-1875-1.1-c3-0-71
Degree $2$
Conductor $1875$
Sign $1$
Analytic cond. $110.628$
Root an. cond. $10.5180$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.24·2-s − 3·3-s + 10.0·4-s + 12.7·6-s + 28.2·7-s − 8.73·8-s + 9·9-s + 51.1·11-s − 30.1·12-s + 25.6·13-s − 120.·14-s − 43.3·16-s − 93.3·17-s − 38.2·18-s − 117.·19-s − 84.8·21-s − 217.·22-s + 94.4·23-s + 26.2·24-s − 109.·26-s − 27·27-s + 284.·28-s + 57.7·29-s − 40.5·31-s + 253.·32-s − 153.·33-s + 396.·34-s + ⋯
L(s)  = 1  − 1.50·2-s − 0.577·3-s + 1.25·4-s + 0.867·6-s + 1.52·7-s − 0.386·8-s + 0.333·9-s + 1.40·11-s − 0.725·12-s + 0.547·13-s − 2.29·14-s − 0.676·16-s − 1.33·17-s − 0.500·18-s − 1.42·19-s − 0.881·21-s − 2.10·22-s + 0.856·23-s + 0.222·24-s − 0.822·26-s − 0.192·27-s + 1.91·28-s + 0.369·29-s − 0.234·31-s + 1.40·32-s − 0.809·33-s + 2.00·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1875\)    =    \(3 \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(110.628\)
Root analytic conductor: \(10.5180\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1875,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.110305785\)
\(L(\frac12)\) \(\approx\) \(1.110305785\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 3T \)
5 \( 1 \)
good2 \( 1 + 4.24T + 8T^{2} \)
7 \( 1 - 28.2T + 343T^{2} \)
11 \( 1 - 51.1T + 1.33e3T^{2} \)
13 \( 1 - 25.6T + 2.19e3T^{2} \)
17 \( 1 + 93.3T + 4.91e3T^{2} \)
19 \( 1 + 117.T + 6.85e3T^{2} \)
23 \( 1 - 94.4T + 1.21e4T^{2} \)
29 \( 1 - 57.7T + 2.43e4T^{2} \)
31 \( 1 + 40.5T + 2.97e4T^{2} \)
37 \( 1 - 409.T + 5.06e4T^{2} \)
41 \( 1 - 101.T + 6.89e4T^{2} \)
43 \( 1 - 1.56T + 7.95e4T^{2} \)
47 \( 1 - 328.T + 1.03e5T^{2} \)
53 \( 1 - 455.T + 1.48e5T^{2} \)
59 \( 1 + 389.T + 2.05e5T^{2} \)
61 \( 1 + 483.T + 2.26e5T^{2} \)
67 \( 1 + 151.T + 3.00e5T^{2} \)
71 \( 1 - 196.T + 3.57e5T^{2} \)
73 \( 1 + 602.T + 3.89e5T^{2} \)
79 \( 1 - 598.T + 4.93e5T^{2} \)
83 \( 1 - 808.T + 5.71e5T^{2} \)
89 \( 1 + 652.T + 7.04e5T^{2} \)
97 \( 1 + 1.55e3T + 9.12e5T^{2} \)
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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

−8.853759074446158166522562210821, −8.333831893236840007954961024373, −7.44564003162599485197808420243, −6.69105757311532020305637370951, −6.00692330099567726273722367939, −4.58509370334455740944048880469, −4.23822386404325308080863977157, −2.27975526589073964495350670683, −1.47269984861760278834827757058, −0.70605268349947070597849989155, 0.70605268349947070597849989155, 1.47269984861760278834827757058, 2.27975526589073964495350670683, 4.23822386404325308080863977157, 4.58509370334455740944048880469, 6.00692330099567726273722367939, 6.69105757311532020305637370951, 7.44564003162599485197808420243, 8.333831893236840007954961024373, 8.853759074446158166522562210821

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