Properties

Label 2-1875-1.1-c1-0-35
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.70·2-s − 3-s + 5.32·4-s + 2.70·6-s − 0.470·7-s − 8.99·8-s + 9-s − 3.18·11-s − 5.32·12-s − 0.563·13-s + 1.27·14-s + 13.7·16-s + 1.70·17-s − 2.70·18-s + 3.74·19-s + 0.470·21-s + 8.61·22-s + 2.26·23-s + 8.99·24-s + 1.52·26-s − 27-s − 2.50·28-s − 8.32·29-s + 5.43·31-s − 19.0·32-s + 3.18·33-s − 4.61·34-s + ⋯
L(s)  = 1  − 1.91·2-s − 0.577·3-s + 2.66·4-s + 1.10·6-s − 0.177·7-s − 3.18·8-s + 0.333·9-s − 0.959·11-s − 1.53·12-s − 0.156·13-s + 0.340·14-s + 3.42·16-s + 0.413·17-s − 0.637·18-s + 0.858·19-s + 0.102·21-s + 1.83·22-s + 0.473·23-s + 1.83·24-s + 0.299·26-s − 0.192·27-s − 0.473·28-s − 1.54·29-s + 0.976·31-s − 3.37·32-s + 0.553·33-s − 0.791·34-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: \(1\)
Selberg data: \((2,\ 1875,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.70T + 2T^{2} \)
7 \( 1 + 0.470T + 7T^{2} \)
11 \( 1 + 3.18T + 11T^{2} \)
13 \( 1 + 0.563T + 13T^{2} \)
17 \( 1 - 1.70T + 17T^{2} \)
19 \( 1 - 3.74T + 19T^{2} \)
23 \( 1 - 2.26T + 23T^{2} \)
29 \( 1 + 8.32T + 29T^{2} \)
31 \( 1 - 5.43T + 31T^{2} \)
37 \( 1 + 1.02T + 37T^{2} \)
41 \( 1 - 1.47T + 41T^{2} \)
43 \( 1 + 6.72T + 43T^{2} \)
47 \( 1 - 4.43T + 47T^{2} \)
53 \( 1 - 7.05T + 53T^{2} \)
59 \( 1 + 12.8T + 59T^{2} \)
61 \( 1 - 0.126T + 61T^{2} \)
67 \( 1 - 2.79T + 67T^{2} \)
71 \( 1 - 16.0T + 71T^{2} \)
73 \( 1 - 9.78T + 73T^{2} \)
79 \( 1 + 4.75T + 79T^{2} \)
83 \( 1 + 11.6T + 83T^{2} \)
89 \( 1 + 6.20T + 89T^{2} \)
97 \( 1 + 8.45T + 97T^{2} \)
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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

−8.920271075899857270774857007475, −8.026302208785476983504198356093, −7.49386559531172778420313925340, −6.79041910701099933218731601384, −5.89268101871923316925531281612, −5.10450689436898656332787697746, −3.38472164127444139805437488134, −2.38139094590099704251695070576, −1.19850140155590226581111780388, 0, 1.19850140155590226581111780388, 2.38139094590099704251695070576, 3.38472164127444139805437488134, 5.10450689436898656332787697746, 5.89268101871923316925531281612, 6.79041910701099933218731601384, 7.49386559531172778420313925340, 8.026302208785476983504198356093, 8.920271075899857270774857007475

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