Properties

Label 2-1575-1.1-c3-0-103
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  + 0.504·2-s − 7.74·4-s + 7·7-s − 7.94·8-s − 54.8·11-s + 16.0·13-s + 3.53·14-s + 57.9·16-s + 0.422·17-s + 127.·19-s − 27.7·22-s − 51.1·23-s + 8.08·26-s − 54.2·28-s − 41.4·29-s + 192.·31-s + 92.8·32-s + 0.213·34-s − 189.·37-s + 64.3·38-s + 76.3·41-s + 294.·43-s + 425.·44-s − 25.7·46-s − 540.·47-s + 49·49-s − 123.·52-s + ⋯
L(s)  = 1  + 0.178·2-s − 0.968·4-s + 0.377·7-s − 0.351·8-s − 1.50·11-s + 0.341·13-s + 0.0674·14-s + 0.905·16-s + 0.00602·17-s + 1.53·19-s − 0.268·22-s − 0.463·23-s + 0.0609·26-s − 0.365·28-s − 0.265·29-s + 1.11·31-s + 0.512·32-s + 0.00107·34-s − 0.840·37-s + 0.274·38-s + 0.290·41-s + 1.04·43-s + 1.45·44-s − 0.0826·46-s − 1.67·47-s + 0.142·49-s − 0.330·52-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 \( 1 - 0.504T + 8T^{2} \)
11 \( 1 + 54.8T + 1.33e3T^{2} \)
13 \( 1 - 16.0T + 2.19e3T^{2} \)
17 \( 1 - 0.422T + 4.91e3T^{2} \)
19 \( 1 - 127.T + 6.85e3T^{2} \)
23 \( 1 + 51.1T + 1.21e4T^{2} \)
29 \( 1 + 41.4T + 2.43e4T^{2} \)
31 \( 1 - 192.T + 2.97e4T^{2} \)
37 \( 1 + 189.T + 5.06e4T^{2} \)
41 \( 1 - 76.3T + 6.89e4T^{2} \)
43 \( 1 - 294.T + 7.95e4T^{2} \)
47 \( 1 + 540.T + 1.03e5T^{2} \)
53 \( 1 - 661.T + 1.48e5T^{2} \)
59 \( 1 + 410.T + 2.05e5T^{2} \)
61 \( 1 - 46.0T + 2.26e5T^{2} \)
67 \( 1 - 10.4T + 3.00e5T^{2} \)
71 \( 1 - 491.T + 3.57e5T^{2} \)
73 \( 1 + 814.T + 3.89e5T^{2} \)
79 \( 1 + 858.T + 4.93e5T^{2} \)
83 \( 1 - 1.05e3T + 5.71e5T^{2} \)
89 \( 1 + 341.T + 7.04e5T^{2} \)
97 \( 1 + 1.41e3T + 9.12e5T^{2} \)
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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

−8.538877691483014295633354723248, −7.995127227814632421923625035794, −7.25305955713484787817781560238, −5.90348465202619142901548779861, −5.28679625616914132438790769323, −4.60361337773776209181337607797, −3.55528783844966165002727701114, −2.63785556257609362022225206533, −1.16458108812078835901304952419, 0, 1.16458108812078835901304952419, 2.63785556257609362022225206533, 3.55528783844966165002727701114, 4.60361337773776209181337607797, 5.28679625616914132438790769323, 5.90348465202619142901548779861, 7.25305955713484787817781560238, 7.995127227814632421923625035794, 8.538877691483014295633354723248

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