Properties

Label 2-1575-1.1-c3-0-50
Degree $2$
Conductor $1575$
Sign $1$
Analytic cond. $92.9280$
Root an. cond. $9.63991$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.82·2-s − 4.65·4-s + 7·7-s − 23.1·8-s + 64.5·11-s + 32.3·13-s + 12.7·14-s − 5.05·16-s − 56.3·17-s − 2.74·19-s + 118.·22-s + 88.1·23-s + 59.1·26-s − 32.5·28-s − 246.·29-s − 110.·31-s + 175.·32-s − 103.·34-s − 120.·37-s − 5.01·38-s + 176.·41-s + 443.·43-s − 300.·44-s + 161.·46-s − 345.·47-s + 49·49-s − 150.·52-s + ⋯
L(s)  = 1  + 0.646·2-s − 0.582·4-s + 0.377·7-s − 1.02·8-s + 1.76·11-s + 0.690·13-s + 0.244·14-s − 0.0790·16-s − 0.803·17-s − 0.0331·19-s + 1.14·22-s + 0.799·23-s + 0.446·26-s − 0.220·28-s − 1.57·29-s − 0.642·31-s + 0.971·32-s − 0.519·34-s − 0.536·37-s − 0.0214·38-s + 0.671·41-s + 1.57·43-s − 1.03·44-s + 0.516·46-s − 1.07·47-s + 0.142·49-s − 0.401·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
Sign: $1$
Analytic conductor: \(92.9280\)
Root analytic conductor: \(9.63991\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1575} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1575,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.764164022\)
\(L(\frac12)\) \(\approx\) \(2.764164022\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 - 7T \)
good2 \( 1 - 1.82T + 8T^{2} \)
11 \( 1 - 64.5T + 1.33e3T^{2} \)
13 \( 1 - 32.3T + 2.19e3T^{2} \)
17 \( 1 + 56.3T + 4.91e3T^{2} \)
19 \( 1 + 2.74T + 6.85e3T^{2} \)
23 \( 1 - 88.1T + 1.21e4T^{2} \)
29 \( 1 + 246.T + 2.43e4T^{2} \)
31 \( 1 + 110.T + 2.97e4T^{2} \)
37 \( 1 + 120.T + 5.06e4T^{2} \)
41 \( 1 - 176.T + 6.89e4T^{2} \)
43 \( 1 - 443.T + 7.95e4T^{2} \)
47 \( 1 + 345.T + 1.03e5T^{2} \)
53 \( 1 - 260.T + 1.48e5T^{2} \)
59 \( 1 + 628.T + 2.05e5T^{2} \)
61 \( 1 + 115.T + 2.26e5T^{2} \)
67 \( 1 - 951.T + 3.00e5T^{2} \)
71 \( 1 + 356.T + 3.57e5T^{2} \)
73 \( 1 - 656.T + 3.89e5T^{2} \)
79 \( 1 - 440.T + 4.93e5T^{2} \)
83 \( 1 + 54.4T + 5.71e5T^{2} \)
89 \( 1 - 1.01e3T + 7.04e5T^{2} \)
97 \( 1 - 724.T + 9.12e5T^{2} \)
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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

−9.148862085010711906584867969920, −8.484545641305012890080494909306, −7.34273971571518094750399983007, −6.43627307088392360868036088376, −5.76120235415806023984537743350, −4.77491454911165300070185180077, −3.99789702734946826772808536767, −3.41717496467596497901784353303, −1.91194494073805489290818219797, −0.75739553855367824648032229375, 0.75739553855367824648032229375, 1.91194494073805489290818219797, 3.41717496467596497901784353303, 3.99789702734946826772808536767, 4.77491454911165300070185180077, 5.76120235415806023984537743350, 6.43627307088392360868036088376, 7.34273971571518094750399983007, 8.484545641305012890080494909306, 9.148862085010711906584867969920

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