Properties

Label 2-1875-1.1-c3-0-132
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  − 2.33·2-s + 3·3-s − 2.52·4-s − 7.01·6-s − 32.9·7-s + 24.6·8-s + 9·9-s + 49.5·11-s − 7.57·12-s − 51.9·13-s + 77.0·14-s − 37.4·16-s − 59.1·17-s − 21.0·18-s + 146.·19-s − 98.7·21-s − 115.·22-s − 48.6·23-s + 73.8·24-s + 121.·26-s + 27·27-s + 83.2·28-s − 32.7·29-s − 81.7·31-s − 109.·32-s + 148.·33-s + 138.·34-s + ⋯
L(s)  = 1  − 0.827·2-s + 0.577·3-s − 0.315·4-s − 0.477·6-s − 1.77·7-s + 1.08·8-s + 0.333·9-s + 1.35·11-s − 0.182·12-s − 1.10·13-s + 1.47·14-s − 0.584·16-s − 0.843·17-s − 0.275·18-s + 1.76·19-s − 1.02·21-s − 1.12·22-s − 0.440·23-s + 0.628·24-s + 0.917·26-s + 0.192·27-s + 0.561·28-s − 0.209·29-s − 0.473·31-s − 0.604·32-s + 0.783·33-s + 0.697·34-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 \( 1 + 2.33T + 8T^{2} \)
7 \( 1 + 32.9T + 343T^{2} \)
11 \( 1 - 49.5T + 1.33e3T^{2} \)
13 \( 1 + 51.9T + 2.19e3T^{2} \)
17 \( 1 + 59.1T + 4.91e3T^{2} \)
19 \( 1 - 146.T + 6.85e3T^{2} \)
23 \( 1 + 48.6T + 1.21e4T^{2} \)
29 \( 1 + 32.7T + 2.43e4T^{2} \)
31 \( 1 + 81.7T + 2.97e4T^{2} \)
37 \( 1 - 278.T + 5.06e4T^{2} \)
41 \( 1 + 296.T + 6.89e4T^{2} \)
43 \( 1 + 64.8T + 7.95e4T^{2} \)
47 \( 1 - 154.T + 1.03e5T^{2} \)
53 \( 1 - 157.T + 1.48e5T^{2} \)
59 \( 1 - 409.T + 2.05e5T^{2} \)
61 \( 1 + 164.T + 2.26e5T^{2} \)
67 \( 1 + 149.T + 3.00e5T^{2} \)
71 \( 1 - 440.T + 3.57e5T^{2} \)
73 \( 1 - 943.T + 3.89e5T^{2} \)
79 \( 1 - 392.T + 4.93e5T^{2} \)
83 \( 1 + 1.14e3T + 5.71e5T^{2} \)
89 \( 1 - 1.33e3T + 7.04e5T^{2} \)
97 \( 1 + 108.T + 9.12e5T^{2} \)
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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

−8.812884883706747324030768341538, −7.71202327558300369025791030451, −7.06558515796820013564939497273, −6.41863157315424493005622487565, −5.20988768783952541984165822828, −4.08140050144751243130192718757, −3.43526590586306643523125956361, −2.33878725462349952121558559136, −1.03255193792129941695394175201, 0, 1.03255193792129941695394175201, 2.33878725462349952121558559136, 3.43526590586306643523125956361, 4.08140050144751243130192718757, 5.20988768783952541984165822828, 6.41863157315424493005622487565, 7.06558515796820013564939497273, 7.71202327558300369025791030451, 8.812884883706747324030768341538

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