Properties

Label 2-1875-1.1-c3-0-152
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  − 4.01·2-s + 3·3-s + 8.14·4-s − 12.0·6-s + 1.75·7-s − 0.565·8-s + 9·9-s − 23.3·11-s + 24.4·12-s − 53.2·13-s − 7.06·14-s − 62.8·16-s − 20.8·17-s − 36.1·18-s + 18.3·19-s + 5.27·21-s + 93.8·22-s + 43.0·23-s − 1.69·24-s + 214.·26-s + 27·27-s + 14.3·28-s + 133.·29-s + 242.·31-s + 257.·32-s − 70.1·33-s + 83.9·34-s + ⋯
L(s)  = 1  − 1.42·2-s + 0.577·3-s + 1.01·4-s − 0.820·6-s + 0.0949·7-s − 0.0249·8-s + 0.333·9-s − 0.640·11-s + 0.587·12-s − 1.13·13-s − 0.134·14-s − 0.982·16-s − 0.298·17-s − 0.473·18-s + 0.221·19-s + 0.0548·21-s + 0.909·22-s + 0.390·23-s − 0.0144·24-s + 1.61·26-s + 0.192·27-s + 0.0966·28-s + 0.854·29-s + 1.40·31-s + 1.41·32-s − 0.369·33-s + 0.423·34-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 \( 1 + 4.01T + 8T^{2} \)
7 \( 1 - 1.75T + 343T^{2} \)
11 \( 1 + 23.3T + 1.33e3T^{2} \)
13 \( 1 + 53.2T + 2.19e3T^{2} \)
17 \( 1 + 20.8T + 4.91e3T^{2} \)
19 \( 1 - 18.3T + 6.85e3T^{2} \)
23 \( 1 - 43.0T + 1.21e4T^{2} \)
29 \( 1 - 133.T + 2.43e4T^{2} \)
31 \( 1 - 242.T + 2.97e4T^{2} \)
37 \( 1 - 357.T + 5.06e4T^{2} \)
41 \( 1 + 424.T + 6.89e4T^{2} \)
43 \( 1 + 93.5T + 7.95e4T^{2} \)
47 \( 1 + 233.T + 1.03e5T^{2} \)
53 \( 1 + 28.0T + 1.48e5T^{2} \)
59 \( 1 - 451.T + 2.05e5T^{2} \)
61 \( 1 - 861.T + 2.26e5T^{2} \)
67 \( 1 + 621.T + 3.00e5T^{2} \)
71 \( 1 - 944.T + 3.57e5T^{2} \)
73 \( 1 + 687.T + 3.89e5T^{2} \)
79 \( 1 + 560.T + 4.93e5T^{2} \)
83 \( 1 + 941.T + 5.71e5T^{2} \)
89 \( 1 - 1.12e3T + 7.04e5T^{2} \)
97 \( 1 + 1.25e3T + 9.12e5T^{2} \)
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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

−8.299610137938379766091592248045, −8.103352826718793160629229313001, −7.16275819960942072123694490003, −6.56570575846625853088376861931, −5.11991258855494968772251916250, −4.42382540569786453077595268295, −2.95289678907784219660141971165, −2.25030664919541813849766625440, −1.10933655632223883562419887699, 0, 1.10933655632223883562419887699, 2.25030664919541813849766625440, 2.95289678907784219660141971165, 4.42382540569786453077595268295, 5.11991258855494968772251916250, 6.56570575846625853088376861931, 7.16275819960942072123694490003, 8.103352826718793160629229313001, 8.299610137938379766091592248045

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