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Properties

Label 2-1575-15.2-c1-0-14
Degree $2$
Conductor $1575$
Sign $0.374 - 0.927i$
Analytic cond. $12.5764$
Root an. cond. $3.54632$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.366 − 0.366i)2-s + 1.73i·4-s + (0.707 + 0.707i)7-s + (1.36 + 1.36i)8-s + 2.44i·11-s + (3.86 − 3.86i)13-s + 0.517·14-s − 2.46·16-s + (−2 + 2i)17-s − 2i·19-s + (0.896 + 0.896i)22-s + (6.46 + 6.46i)23-s − 2.82i·26-s + (−1.22 + 1.22i)28-s + 6.31·29-s + ⋯
L(s)  = 1  + (0.258 − 0.258i)2-s + 0.866i·4-s + (0.267 + 0.267i)7-s + (0.482 + 0.482i)8-s + 0.738i·11-s + (1.07 − 1.07i)13-s + 0.138·14-s − 0.616·16-s + (−0.485 + 0.485i)17-s − 0.458i·19-s + (0.191 + 0.191i)22-s + (1.34 + 1.34i)23-s − 0.554i·26-s + (−0.231 + 0.231i)28-s + 1.17·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.374 - 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.374 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.374 - 0.927i$
Analytic conductor: \(12.5764\)
Root analytic conductor: \(3.54632\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1575} (1457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1575,\ (\ :1/2),\ 0.374 - 0.927i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.034089603\)
\(L(\frac12)\) \(\approx\) \(2.034089603\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 - 0.707i)T \)
good2 \( 1 + (-0.366 + 0.366i)T - 2iT^{2} \)
11 \( 1 - 2.44iT - 11T^{2} \)
13 \( 1 + (-3.86 + 3.86i)T - 13iT^{2} \)
17 \( 1 + (2 - 2i)T - 17iT^{2} \)
19 \( 1 + 2iT - 19T^{2} \)
23 \( 1 + (-6.46 - 6.46i)T + 23iT^{2} \)
29 \( 1 - 6.31T + 29T^{2} \)
31 \( 1 + 4.92T + 31T^{2} \)
37 \( 1 + (-0.378 - 0.378i)T + 37iT^{2} \)
41 \( 1 - 2.07iT - 41T^{2} \)
43 \( 1 + (-2.82 + 2.82i)T - 43iT^{2} \)
47 \( 1 + (7.46 - 7.46i)T - 47iT^{2} \)
53 \( 1 + (-2.26 - 2.26i)T + 53iT^{2} \)
59 \( 1 + 12.6T + 59T^{2} \)
61 \( 1 + 8.92T + 61T^{2} \)
67 \( 1 + (-6.31 - 6.31i)T + 67iT^{2} \)
71 \( 1 - 4.52iT - 71T^{2} \)
73 \( 1 + (1.03 - 1.03i)T - 73iT^{2} \)
79 \( 1 - 4iT - 79T^{2} \)
83 \( 1 + (-4.53 - 4.53i)T + 83iT^{2} \)
89 \( 1 - 14.1T + 89T^{2} \)
97 \( 1 + (-10.8 - 10.8i)T + 97iT^{2} \)
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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

−9.407951954456889069090678568299, −8.748029693420800607514412268705, −7.971199033425631861761176110106, −7.32984683937574347938413354961, −6.36396376885817724367980431435, −5.28798001453997643096865388196, −4.51299836604256503272810441985, −3.49811738868072556831675113131, −2.74877814542824368579971639039, −1.46643015376985832702736508309, 0.78823675155185782254801107471, 1.92235166855470830278142559391, 3.36133698488011421064372335887, 4.46119237938318921973368429069, 5.04989207492151777447721659129, 6.26543532851890514873621682226, 6.51622308505688525942829597747, 7.54294085637694967837286836983, 8.737175130730861392239947896764, 9.056207888792655299960240541920

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