Properties

Label 2-1575-1.1-c3-0-125
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  + 2.56·2-s − 1.43·4-s + 7·7-s − 24.1·8-s − 6.24·11-s + 56.3·13-s + 17.9·14-s − 50.4·16-s + 24.6·17-s − 90.7·19-s − 16·22-s − 69.8·23-s + 144.·26-s − 10.0·28-s + 228.·29-s − 67.8·31-s + 64.2·32-s + 63.0·34-s + 58.8·37-s − 232.·38-s − 19.2·41-s − 365.·43-s + 8.98·44-s − 178.·46-s − 195.·47-s + 49·49-s − 81.0·52-s + ⋯
L(s)  = 1  + 0.905·2-s − 0.179·4-s + 0.377·7-s − 1.06·8-s − 0.171·11-s + 1.20·13-s + 0.342·14-s − 0.787·16-s + 0.350·17-s − 1.09·19-s − 0.155·22-s − 0.633·23-s + 1.08·26-s − 0.0679·28-s + 1.46·29-s − 0.393·31-s + 0.354·32-s + 0.317·34-s + 0.261·37-s − 0.991·38-s − 0.0731·41-s − 1.29·43-s + 0.0307·44-s − 0.573·46-s − 0.605·47-s + 0.142·49-s − 0.216·52-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 \( 1 - 2.56T + 8T^{2} \)
11 \( 1 + 6.24T + 1.33e3T^{2} \)
13 \( 1 - 56.3T + 2.19e3T^{2} \)
17 \( 1 - 24.6T + 4.91e3T^{2} \)
19 \( 1 + 90.7T + 6.85e3T^{2} \)
23 \( 1 + 69.8T + 1.21e4T^{2} \)
29 \( 1 - 228.T + 2.43e4T^{2} \)
31 \( 1 + 67.8T + 2.97e4T^{2} \)
37 \( 1 - 58.8T + 5.06e4T^{2} \)
41 \( 1 + 19.2T + 6.89e4T^{2} \)
43 \( 1 + 365.T + 7.95e4T^{2} \)
47 \( 1 + 195.T + 1.03e5T^{2} \)
53 \( 1 + 511.T + 1.48e5T^{2} \)
59 \( 1 - 284T + 2.05e5T^{2} \)
61 \( 1 + 123.T + 2.26e5T^{2} \)
67 \( 1 + 144.T + 3.00e5T^{2} \)
71 \( 1 - 73.0T + 3.57e5T^{2} \)
73 \( 1 + 638.T + 3.89e5T^{2} \)
79 \( 1 - 976.T + 4.93e5T^{2} \)
83 \( 1 + 484.T + 5.71e5T^{2} \)
89 \( 1 + 1.01e3T + 7.04e5T^{2} \)
97 \( 1 + 1.80e3T + 9.12e5T^{2} \)
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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

−8.447434868479384988725691843532, −8.147036793056954559523426040963, −6.70172929150530649428914671511, −6.11353470709332543919470748094, −5.25216445786932463049025752119, −4.43223108983375048192022938937, −3.71311493202852684103394189495, −2.75405815060199926597406437818, −1.42982863855845461109290695046, 0, 1.42982863855845461109290695046, 2.75405815060199926597406437818, 3.71311493202852684103394189495, 4.43223108983375048192022938937, 5.25216445786932463049025752119, 6.11353470709332543919470748094, 6.70172929150530649428914671511, 8.147036793056954559523426040963, 8.447434868479384988725691843532

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