Properties

Label 2-1875-1.1-c3-0-176
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.955·2-s + 3·3-s − 7.08·4-s − 2.86·6-s + 12.4·7-s + 14.4·8-s + 9·9-s − 44.7·11-s − 21.2·12-s − 7.13·13-s − 11.9·14-s + 42.9·16-s + 26.5·17-s − 8.59·18-s − 46.3·19-s + 37.4·21-s + 42.7·22-s + 145.·23-s + 43.2·24-s + 6.81·26-s + 27·27-s − 88.4·28-s − 213.·29-s − 148.·31-s − 156.·32-s − 134.·33-s − 25.3·34-s + ⋯
L(s)  = 1  − 0.337·2-s + 0.577·3-s − 0.885·4-s − 0.194·6-s + 0.674·7-s + 0.636·8-s + 0.333·9-s − 1.22·11-s − 0.511·12-s − 0.152·13-s − 0.227·14-s + 0.670·16-s + 0.378·17-s − 0.112·18-s − 0.560·19-s + 0.389·21-s + 0.414·22-s + 1.31·23-s + 0.367·24-s + 0.0514·26-s + 0.192·27-s − 0.597·28-s − 1.36·29-s − 0.857·31-s − 0.863·32-s − 0.707·33-s − 0.127·34-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: $\chi_{1875} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1875,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 + 0.955T + 8T^{2} \)
7 \( 1 - 12.4T + 343T^{2} \)
11 \( 1 + 44.7T + 1.33e3T^{2} \)
13 \( 1 + 7.13T + 2.19e3T^{2} \)
17 \( 1 - 26.5T + 4.91e3T^{2} \)
19 \( 1 + 46.3T + 6.85e3T^{2} \)
23 \( 1 - 145.T + 1.21e4T^{2} \)
29 \( 1 + 213.T + 2.43e4T^{2} \)
31 \( 1 + 148.T + 2.97e4T^{2} \)
37 \( 1 - 402.T + 5.06e4T^{2} \)
41 \( 1 - 233.T + 6.89e4T^{2} \)
43 \( 1 - 87.5T + 7.95e4T^{2} \)
47 \( 1 + 75.9T + 1.03e5T^{2} \)
53 \( 1 - 167.T + 1.48e5T^{2} \)
59 \( 1 + 595.T + 2.05e5T^{2} \)
61 \( 1 + 865.T + 2.26e5T^{2} \)
67 \( 1 + 406.T + 3.00e5T^{2} \)
71 \( 1 - 526.T + 3.57e5T^{2} \)
73 \( 1 - 1.00e3T + 3.89e5T^{2} \)
79 \( 1 + 133.T + 4.93e5T^{2} \)
83 \( 1 - 516.T + 5.71e5T^{2} \)
89 \( 1 + 610.T + 7.04e5T^{2} \)
97 \( 1 + 1.63e3T + 9.12e5T^{2} \)
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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

−8.473847755275551934625609343232, −7.73734914657915793767202827619, −7.42682534209224010969179087657, −5.92591460741424075959775470368, −5.05744788121354316047726591052, −4.45075640258667886909122557228, −3.40250064420180933537486822875, −2.35352474780712372888292473109, −1.22031648274512305475006836792, 0, 1.22031648274512305475006836792, 2.35352474780712372888292473109, 3.40250064420180933537486822875, 4.45075640258667886909122557228, 5.05744788121354316047726591052, 5.92591460741424075959775470368, 7.42682534209224010969179087657, 7.73734914657915793767202827619, 8.473847755275551934625609343232

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