Properties

Label 2-1575-1.1-c3-0-105
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.561·2-s − 7.68·4-s + 7·7-s + 8.80·8-s + 25.8·11-s + 15.5·13-s − 3.93·14-s + 56.5·16-s − 95.1·17-s − 143.·19-s − 14.4·22-s + 77.5·23-s − 8.73·26-s − 53.7·28-s + 204.·29-s − 40.6·31-s − 102.·32-s + 53.4·34-s − 95.6·37-s + 80.5·38-s − 24.7·41-s − 282.·43-s − 198.·44-s − 43.5·46-s + 257.·47-s + 49·49-s − 119.·52-s + ⋯
L(s)  = 1  − 0.198·2-s − 0.960·4-s + 0.377·7-s + 0.389·8-s + 0.707·11-s + 0.331·13-s − 0.0750·14-s + 0.883·16-s − 1.35·17-s − 1.73·19-s − 0.140·22-s + 0.702·23-s − 0.0658·26-s − 0.363·28-s + 1.30·29-s − 0.235·31-s − 0.564·32-s + 0.269·34-s − 0.425·37-s + 0.343·38-s − 0.0941·41-s − 1.00·43-s − 0.679·44-s − 0.139·46-s + 0.798·47-s + 0.142·49-s − 0.318·52-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: \(1\)
Selberg data: \((2,\ 1575,\ (\ :3/2),\ -1)\)

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 \( 1 \)
5 \( 1 \)
7 \( 1 - 7T \)
good2 \( 1 + 0.561T + 8T^{2} \)
11 \( 1 - 25.8T + 1.33e3T^{2} \)
13 \( 1 - 15.5T + 2.19e3T^{2} \)
17 \( 1 + 95.1T + 4.91e3T^{2} \)
19 \( 1 + 143.T + 6.85e3T^{2} \)
23 \( 1 - 77.5T + 1.21e4T^{2} \)
29 \( 1 - 204.T + 2.43e4T^{2} \)
31 \( 1 + 40.6T + 2.97e4T^{2} \)
37 \( 1 + 95.6T + 5.06e4T^{2} \)
41 \( 1 + 24.7T + 6.89e4T^{2} \)
43 \( 1 + 282.T + 7.95e4T^{2} \)
47 \( 1 - 257.T + 1.03e5T^{2} \)
53 \( 1 - 257.T + 1.48e5T^{2} \)
59 \( 1 - 651.T + 2.05e5T^{2} \)
61 \( 1 - 451.T + 2.26e5T^{2} \)
67 \( 1 + 832.T + 3.00e5T^{2} \)
71 \( 1 - 174.T + 3.57e5T^{2} \)
73 \( 1 - 47.4T + 3.89e5T^{2} \)
79 \( 1 + 1.16T + 4.93e5T^{2} \)
83 \( 1 - 1.49e3T + 5.71e5T^{2} \)
89 \( 1 + 1.30e3T + 7.04e5T^{2} \)
97 \( 1 - 1.36e3T + 9.12e5T^{2} \)
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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

−8.689257424532438980298814428305, −8.230175594864463793772414311778, −6.97305647109596391661098159589, −6.34234235654288878296059844512, −5.18093353796941387762300031023, −4.40788446758935621230015993616, −3.79294837469130521737204776784, −2.34758180111414412280306954723, −1.17605116690899426035567815458, 0, 1.17605116690899426035567815458, 2.34758180111414412280306954723, 3.79294837469130521737204776784, 4.40788446758935621230015993616, 5.18093353796941387762300031023, 6.34234235654288878296059844512, 6.97305647109596391661098159589, 8.230175594864463793772414311778, 8.689257424532438980298814428305

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