Properties

Label 2-1575-1.1-c3-0-35
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.58·2-s − 1.31·4-s + 7·7-s − 24.0·8-s − 38.2·11-s − 19.3·13-s + 18.1·14-s − 51.7·16-s − 87.2·17-s − 44.2·19-s − 98.9·22-s + 218.·23-s − 50.0·26-s − 9.19·28-s + 46.9·29-s + 194.·31-s + 58.8·32-s − 225.·34-s − 366.·37-s − 114.·38-s + 339.·41-s + 226.·43-s + 50.2·44-s + 564.·46-s + 11.6·47-s + 49·49-s + 25.4·52-s + ⋯
L(s)  = 1  + 0.914·2-s − 0.164·4-s + 0.377·7-s − 1.06·8-s − 1.04·11-s − 0.412·13-s + 0.345·14-s − 0.808·16-s − 1.24·17-s − 0.534·19-s − 0.958·22-s + 1.97·23-s − 0.377·26-s − 0.0620·28-s + 0.300·29-s + 1.12·31-s + 0.324·32-s − 1.13·34-s − 1.63·37-s − 0.488·38-s + 1.29·41-s + 0.802·43-s + 0.172·44-s + 1.80·46-s + 0.0362·47-s + 0.142·49-s + 0.0677·52-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: \(0\)
Selberg data: \((2,\ 1575,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.196711309\)
\(L(\frac12)\) \(\approx\) \(2.196711309\)
\(L(\frac{5}{2})\) 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 \( 1 - 2.58T + 8T^{2} \)
11 \( 1 + 38.2T + 1.33e3T^{2} \)
13 \( 1 + 19.3T + 2.19e3T^{2} \)
17 \( 1 + 87.2T + 4.91e3T^{2} \)
19 \( 1 + 44.2T + 6.85e3T^{2} \)
23 \( 1 - 218.T + 1.21e4T^{2} \)
29 \( 1 - 46.9T + 2.43e4T^{2} \)
31 \( 1 - 194.T + 2.97e4T^{2} \)
37 \( 1 + 366.T + 5.06e4T^{2} \)
41 \( 1 - 339.T + 6.89e4T^{2} \)
43 \( 1 - 226.T + 7.95e4T^{2} \)
47 \( 1 - 11.6T + 1.03e5T^{2} \)
53 \( 1 + 209.T + 1.48e5T^{2} \)
59 \( 1 - 616T + 2.05e5T^{2} \)
61 \( 1 - 320.T + 2.26e5T^{2} \)
67 \( 1 + 14.5T + 3.00e5T^{2} \)
71 \( 1 - 952T + 3.57e5T^{2} \)
73 \( 1 + 824.T + 3.89e5T^{2} \)
79 \( 1 - 156.T + 4.93e5T^{2} \)
83 \( 1 + 1.03e3T + 5.71e5T^{2} \)
89 \( 1 - 170.T + 7.04e5T^{2} \)
97 \( 1 + 1.05e3T + 9.12e5T^{2} \)
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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

−8.881982939758830979889589943622, −8.437540764171217802569233428049, −7.29298981146451621725259548094, −6.53512913539523144555875267523, −5.49208933703600007064020239737, −4.86421566479748640088953277247, −4.25417421365885802669837411108, −3.03848523496001632872775020048, −2.30809900767388224883449471779, −0.60335729966748845531091913640, 0.60335729966748845531091913640, 2.30809900767388224883449471779, 3.03848523496001632872775020048, 4.25417421365885802669837411108, 4.86421566479748640088953277247, 5.49208933703600007064020239737, 6.53512913539523144555875267523, 7.29298981146451621725259548094, 8.437540764171217802569233428049, 8.881982939758830979889589943622

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