Properties

Label 2-1575-1.1-c3-0-58
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  − 3.70·2-s + 5.70·4-s + 7·7-s + 8.50·8-s + 60.0·11-s + 0.387·13-s − 25.9·14-s − 77.1·16-s + 35.4·17-s − 6.08·19-s − 222.·22-s + 31.5·23-s − 1.43·26-s + 39.9·28-s + 292.·29-s + 130.·31-s + 217.·32-s − 131.·34-s + 219.·37-s + 22.5·38-s + 447.·41-s + 210.·43-s + 342.·44-s − 116.·46-s − 457.·47-s + 49·49-s + 2.20·52-s + ⋯
L(s)  = 1  − 1.30·2-s + 0.712·4-s + 0.377·7-s + 0.375·8-s + 1.64·11-s + 0.00826·13-s − 0.494·14-s − 1.20·16-s + 0.506·17-s − 0.0735·19-s − 2.15·22-s + 0.285·23-s − 0.0108·26-s + 0.269·28-s + 1.87·29-s + 0.754·31-s + 1.20·32-s − 0.662·34-s + 0.977·37-s + 0.0962·38-s + 1.70·41-s + 0.744·43-s + 1.17·44-s − 0.373·46-s − 1.42·47-s + 0.142·49-s + 0.00589·52-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

\(L(2)\) \(\approx\) \(1.429065583\)
\(L(\frac12)\) \(\approx\) \(1.429065583\)
\(L(\frac{5}{2})\) 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 \( 1 + 3.70T + 8T^{2} \)
11 \( 1 - 60.0T + 1.33e3T^{2} \)
13 \( 1 - 0.387T + 2.19e3T^{2} \)
17 \( 1 - 35.4T + 4.91e3T^{2} \)
19 \( 1 + 6.08T + 6.85e3T^{2} \)
23 \( 1 - 31.5T + 1.21e4T^{2} \)
29 \( 1 - 292.T + 2.43e4T^{2} \)
31 \( 1 - 130.T + 2.97e4T^{2} \)
37 \( 1 - 219.T + 5.06e4T^{2} \)
41 \( 1 - 447.T + 6.89e4T^{2} \)
43 \( 1 - 210.T + 7.95e4T^{2} \)
47 \( 1 + 457.T + 1.03e5T^{2} \)
53 \( 1 + 144.T + 1.48e5T^{2} \)
59 \( 1 + 767.T + 2.05e5T^{2} \)
61 \( 1 - 667.T + 2.26e5T^{2} \)
67 \( 1 + 77.4T + 3.00e5T^{2} \)
71 \( 1 - 906.T + 3.57e5T^{2} \)
73 \( 1 + 1.02e3T + 3.89e5T^{2} \)
79 \( 1 + 690.T + 4.93e5T^{2} \)
83 \( 1 - 979.T + 5.71e5T^{2} \)
89 \( 1 - 910.T + 7.04e5T^{2} \)
97 \( 1 - 11.1T + 9.12e5T^{2} \)
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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

−9.156103236349722025099759684744, −8.325848433073911748200070429121, −7.76223597770825459068182732620, −6.78598713248978794164769665766, −6.17398822645377189219624232740, −4.78176142832046828881042031172, −4.06921069039322911005732396761, −2.70279145089734958656199450452, −1.40869747973902791101569983002, −0.818406656170659476356281396700, 0.818406656170659476356281396700, 1.40869747973902791101569983002, 2.70279145089734958656199450452, 4.06921069039322911005732396761, 4.78176142832046828881042031172, 6.17398822645377189219624232740, 6.78598713248978794164769665766, 7.76223597770825459068182732620, 8.325848433073911748200070429121, 9.156103236349722025099759684744

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