Properties

Label 2-1875-1.1-c1-0-61
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  − 1.82·2-s + 3-s + 1.33·4-s − 1.82·6-s + 1.44·7-s + 1.20·8-s + 9-s − 2.12·11-s + 1.33·12-s − 5.70·13-s − 2.64·14-s − 4.88·16-s + 4.15·17-s − 1.82·18-s + 1.70·19-s + 1.44·21-s + 3.89·22-s − 0.323·23-s + 1.20·24-s + 10.4·26-s + 27-s + 1.93·28-s − 8.74·29-s − 8.45·31-s + 6.50·32-s − 2.12·33-s − 7.59·34-s + ⋯
L(s)  = 1  − 1.29·2-s + 0.577·3-s + 0.669·4-s − 0.745·6-s + 0.546·7-s + 0.427·8-s + 0.333·9-s − 0.641·11-s + 0.386·12-s − 1.58·13-s − 0.705·14-s − 1.22·16-s + 1.00·17-s − 0.430·18-s + 0.390·19-s + 0.315·21-s + 0.829·22-s − 0.0674·23-s + 0.246·24-s + 2.04·26-s + 0.192·27-s + 0.365·28-s − 1.62·29-s − 1.51·31-s + 1.15·32-s − 0.370·33-s − 1.30·34-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 \( 1 + 1.82T + 2T^{2} \)
7 \( 1 - 1.44T + 7T^{2} \)
11 \( 1 + 2.12T + 11T^{2} \)
13 \( 1 + 5.70T + 13T^{2} \)
17 \( 1 - 4.15T + 17T^{2} \)
19 \( 1 - 1.70T + 19T^{2} \)
23 \( 1 + 0.323T + 23T^{2} \)
29 \( 1 + 8.74T + 29T^{2} \)
31 \( 1 + 8.45T + 31T^{2} \)
37 \( 1 - 1.75T + 37T^{2} \)
41 \( 1 + 6.87T + 41T^{2} \)
43 \( 1 + 11.1T + 43T^{2} \)
47 \( 1 - 12.5T + 47T^{2} \)
53 \( 1 - 8.34T + 53T^{2} \)
59 \( 1 + 2.12T + 59T^{2} \)
61 \( 1 + 5.38T + 61T^{2} \)
67 \( 1 + 7.13T + 67T^{2} \)
71 \( 1 - 2.67T + 71T^{2} \)
73 \( 1 - 6.28T + 73T^{2} \)
79 \( 1 - 8.37T + 79T^{2} \)
83 \( 1 + 14.5T + 83T^{2} \)
89 \( 1 - 2.68T + 89T^{2} \)
97 \( 1 - 8.55T + 97T^{2} \)
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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

−8.905767466265111573255116842723, −8.021482881025705419872424519872, −7.53203692575450239717073009276, −7.09106401508749150117503045913, −5.49587397555262134081745935286, −4.85362078670601144020044782698, −3.63432428239703359984648049004, −2.41596493930272092690263645912, −1.56517115121248049417204700700, 0, 1.56517115121248049417204700700, 2.41596493930272092690263645912, 3.63432428239703359984648049004, 4.85362078670601144020044782698, 5.49587397555262134081745935286, 7.09106401508749150117503045913, 7.53203692575450239717073009276, 8.021482881025705419872424519872, 8.905767466265111573255116842723

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