Properties

Label 2-1575-1.1-c3-0-94
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  − 1.56·2-s − 5.56·4-s + 7·7-s + 21.1·8-s + 10.2·11-s − 34.3·13-s − 10.9·14-s + 11.4·16-s − 82.6·17-s + 90.7·19-s − 16·22-s − 12.1·23-s + 53.6·26-s − 38.9·28-s + 105.·29-s − 142.·31-s − 187.·32-s + 128.·34-s − 64.8·37-s − 141.·38-s + 195.·41-s + 319.·43-s − 56.9·44-s + 18.9·46-s − 318.·47-s + 49·49-s + 191.·52-s + ⋯
L(s)  = 1  − 0.552·2-s − 0.695·4-s + 0.377·7-s + 0.935·8-s + 0.280·11-s − 0.732·13-s − 0.208·14-s + 0.178·16-s − 1.17·17-s + 1.09·19-s − 0.155·22-s − 0.110·23-s + 0.404·26-s − 0.262·28-s + 0.673·29-s − 0.823·31-s − 1.03·32-s + 0.650·34-s − 0.288·37-s − 0.604·38-s + 0.743·41-s + 1.13·43-s − 0.195·44-s + 0.0607·46-s − 0.989·47-s + 0.142·49-s + 0.509·52-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 \( 1 + 1.56T + 8T^{2} \)
11 \( 1 - 10.2T + 1.33e3T^{2} \)
13 \( 1 + 34.3T + 2.19e3T^{2} \)
17 \( 1 + 82.6T + 4.91e3T^{2} \)
19 \( 1 - 90.7T + 6.85e3T^{2} \)
23 \( 1 + 12.1T + 1.21e4T^{2} \)
29 \( 1 - 105.T + 2.43e4T^{2} \)
31 \( 1 + 142.T + 2.97e4T^{2} \)
37 \( 1 + 64.8T + 5.06e4T^{2} \)
41 \( 1 - 195.T + 6.89e4T^{2} \)
43 \( 1 - 319.T + 7.95e4T^{2} \)
47 \( 1 + 318.T + 1.03e5T^{2} \)
53 \( 1 + 296.T + 1.48e5T^{2} \)
59 \( 1 - 284T + 2.05e5T^{2} \)
61 \( 1 + 494.T + 2.26e5T^{2} \)
67 \( 1 + 549.T + 3.00e5T^{2} \)
71 \( 1 - 740.T + 3.57e5T^{2} \)
73 \( 1 - 556.T + 3.89e5T^{2} \)
79 \( 1 + 376.T + 4.93e5T^{2} \)
83 \( 1 - 752.T + 5.71e5T^{2} \)
89 \( 1 - 945.T + 7.04e5T^{2} \)
97 \( 1 - 180.T + 9.12e5T^{2} \)
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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

−8.868738706159550935696963911808, −7.86596886154450135500752780172, −7.35832138870063003989500672696, −6.29938672590132057834484863921, −5.14565052382689317365983629469, −4.59449175593685201203137836104, −3.59802994012841452569613876974, −2.27167902505665643537899197277, −1.12816888764171786518160287592, 0, 1.12816888764171786518160287592, 2.27167902505665643537899197277, 3.59802994012841452569613876974, 4.59449175593685201203137836104, 5.14565052382689317365983629469, 6.29938672590132057834484863921, 7.35832138870063003989500672696, 7.86596886154450135500752780172, 8.868738706159550935696963911808

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