Properties

Label 2-1875-1.1-c3-0-158
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  − 0.574·2-s + 3·3-s − 7.66·4-s − 1.72·6-s − 2.67·7-s + 9.00·8-s + 9·9-s − 63.0·11-s − 23.0·12-s + 14.1·13-s + 1.53·14-s + 56.1·16-s + 30.3·17-s − 5.17·18-s + 27.5·19-s − 8.03·21-s + 36.2·22-s + 116.·23-s + 27.0·24-s − 8.14·26-s + 27·27-s + 20.5·28-s − 40.2·29-s − 222.·31-s − 104.·32-s − 189.·33-s − 17.4·34-s + ⋯
L(s)  = 1  − 0.203·2-s + 0.577·3-s − 0.958·4-s − 0.117·6-s − 0.144·7-s + 0.397·8-s + 0.333·9-s − 1.72·11-s − 0.553·12-s + 0.302·13-s + 0.0293·14-s + 0.877·16-s + 0.433·17-s − 0.0677·18-s + 0.332·19-s − 0.0834·21-s + 0.351·22-s + 1.05·23-s + 0.229·24-s − 0.0614·26-s + 0.192·27-s + 0.138·28-s − 0.257·29-s − 1.29·31-s − 0.576·32-s − 0.997·33-s − 0.0880·34-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: \(1\)
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 \( 1 + 0.574T + 8T^{2} \)
7 \( 1 + 2.67T + 343T^{2} \)
11 \( 1 + 63.0T + 1.33e3T^{2} \)
13 \( 1 - 14.1T + 2.19e3T^{2} \)
17 \( 1 - 30.3T + 4.91e3T^{2} \)
19 \( 1 - 27.5T + 6.85e3T^{2} \)
23 \( 1 - 116.T + 1.21e4T^{2} \)
29 \( 1 + 40.2T + 2.43e4T^{2} \)
31 \( 1 + 222.T + 2.97e4T^{2} \)
37 \( 1 - 82.8T + 5.06e4T^{2} \)
41 \( 1 - 144.T + 6.89e4T^{2} \)
43 \( 1 - 433.T + 7.95e4T^{2} \)
47 \( 1 - 459.T + 1.03e5T^{2} \)
53 \( 1 + 247.T + 1.48e5T^{2} \)
59 \( 1 - 310.T + 2.05e5T^{2} \)
61 \( 1 - 2.08T + 2.26e5T^{2} \)
67 \( 1 - 654.T + 3.00e5T^{2} \)
71 \( 1 + 908.T + 3.57e5T^{2} \)
73 \( 1 + 878.T + 3.89e5T^{2} \)
79 \( 1 + 1.13e3T + 4.93e5T^{2} \)
83 \( 1 + 986.T + 5.71e5T^{2} \)
89 \( 1 - 988.T + 7.04e5T^{2} \)
97 \( 1 - 1.53e3T + 9.12e5T^{2} \)
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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

−8.614029251687486940060394603885, −7.64206287520019103175514099444, −7.39975747002800178273995363981, −5.83539819596070788372657650806, −5.22763938163780673158771855999, −4.32966512503761102178590934299, −3.36396088703050791094045097434, −2.53298457950116853387509092200, −1.14416233404640394625026866907, 0, 1.14416233404640394625026866907, 2.53298457950116853387509092200, 3.36396088703050791094045097434, 4.32966512503761102178590934299, 5.22763938163780673158771855999, 5.83539819596070788372657650806, 7.39975747002800178273995363981, 7.64206287520019103175514099444, 8.614029251687486940060394603885

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