Properties

Label 2-1575-1.1-c3-0-96
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  − 3.33·2-s + 3.14·4-s + 7·7-s + 16.2·8-s − 18.3·11-s + 10.1·13-s − 23.3·14-s − 79.2·16-s + 24.6·17-s + 77.4·19-s + 61.1·22-s − 149.·23-s − 33.8·26-s + 21.9·28-s + 10.2·29-s + 124.·31-s + 134.·32-s − 82.4·34-s + 215.·37-s − 258.·38-s − 495.·41-s − 220.·43-s − 57.5·44-s + 498.·46-s − 212.·47-s + 49·49-s + 31.8·52-s + ⋯
L(s)  = 1  − 1.18·2-s + 0.392·4-s + 0.377·7-s + 0.716·8-s − 0.502·11-s + 0.216·13-s − 0.446·14-s − 1.23·16-s + 0.352·17-s + 0.934·19-s + 0.592·22-s − 1.35·23-s − 0.255·26-s + 0.148·28-s + 0.0659·29-s + 0.720·31-s + 0.744·32-s − 0.415·34-s + 0.959·37-s − 1.10·38-s − 1.88·41-s − 0.782·43-s − 0.197·44-s + 1.59·46-s − 0.660·47-s + 0.142·49-s + 0.0849·52-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: Trivial
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 \( 1 + 3.33T + 8T^{2} \)
11 \( 1 + 18.3T + 1.33e3T^{2} \)
13 \( 1 - 10.1T + 2.19e3T^{2} \)
17 \( 1 - 24.6T + 4.91e3T^{2} \)
19 \( 1 - 77.4T + 6.85e3T^{2} \)
23 \( 1 + 149.T + 1.21e4T^{2} \)
29 \( 1 - 10.2T + 2.43e4T^{2} \)
31 \( 1 - 124.T + 2.97e4T^{2} \)
37 \( 1 - 215.T + 5.06e4T^{2} \)
41 \( 1 + 495.T + 6.89e4T^{2} \)
43 \( 1 + 220.T + 7.95e4T^{2} \)
47 \( 1 + 212.T + 1.03e5T^{2} \)
53 \( 1 + 532.T + 1.48e5T^{2} \)
59 \( 1 - 324.T + 2.05e5T^{2} \)
61 \( 1 - 653.T + 2.26e5T^{2} \)
67 \( 1 + 819.T + 3.00e5T^{2} \)
71 \( 1 - 466.T + 3.57e5T^{2} \)
73 \( 1 - 173.T + 3.89e5T^{2} \)
79 \( 1 - 810.T + 4.93e5T^{2} \)
83 \( 1 + 12.3T + 5.71e5T^{2} \)
89 \( 1 - 33.8T + 7.04e5T^{2} \)
97 \( 1 + 810.T + 9.12e5T^{2} \)
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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

−8.431302409552844887048218092715, −8.155822130649367302019163188763, −7.38290793853055917085704931505, −6.44747609415606882227317165859, −5.34986597705328759951354143749, −4.54755104117758441602481240982, −3.38533794506503599883149516020, −2.09112442625158890914985815567, −1.13210883522260627177943633406, 0, 1.13210883522260627177943633406, 2.09112442625158890914985815567, 3.38533794506503599883149516020, 4.54755104117758441602481240982, 5.34986597705328759951354143749, 6.44747609415606882227317165859, 7.38290793853055917085704931505, 8.155822130649367302019163188763, 8.431302409552844887048218092715

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