Properties

Degree $2$
Conductor $525$
Sign $1$
Motivic weight $3$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 3·3-s − 4·4-s − 6·6-s + 7·7-s − 24·8-s + 9·9-s − 21·11-s + 12·12-s − 24·13-s + 14·14-s − 16·16-s + 22·17-s + 18·18-s + 16·19-s − 21·21-s − 42·22-s + 25·23-s + 72·24-s − 48·26-s − 27·27-s − 28·28-s + 167·29-s + 10·31-s + 160·32-s + 63·33-s + 44·34-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.408·6-s + 0.377·7-s − 1.06·8-s + 1/3·9-s − 0.575·11-s + 0.288·12-s − 0.512·13-s + 0.267·14-s − 1/4·16-s + 0.313·17-s + 0.235·18-s + 0.193·19-s − 0.218·21-s − 0.407·22-s + 0.226·23-s + 0.612·24-s − 0.362·26-s − 0.192·27-s − 0.188·28-s + 1.06·29-s + 0.0579·31-s + 0.883·32-s + 0.332·33-s + 0.221·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $1$
Motivic weight: \(3\)
Character: $\chi_{525} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.642064131\)
\(L(\frac12)\) \(\approx\) \(1.642064131\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + p T \)
5 \( 1 \)
7 \( 1 - p T \)
good2 \( 1 - p T + p^{3} T^{2} \)
11 \( 1 + 21 T + p^{3} T^{2} \)
13 \( 1 + 24 T + p^{3} T^{2} \)
17 \( 1 - 22 T + p^{3} T^{2} \)
19 \( 1 - 16 T + p^{3} T^{2} \)
23 \( 1 - 25 T + p^{3} T^{2} \)
29 \( 1 - 167 T + p^{3} T^{2} \)
31 \( 1 - 10 T + p^{3} T^{2} \)
37 \( 1 - 133 T + p^{3} T^{2} \)
41 \( 1 + 168 T + p^{3} T^{2} \)
43 \( 1 - 97 T + p^{3} T^{2} \)
47 \( 1 - 400 T + p^{3} T^{2} \)
53 \( 1 - 182 T + p^{3} T^{2} \)
59 \( 1 - 488 T + p^{3} T^{2} \)
61 \( 1 - 28 T + p^{3} T^{2} \)
67 \( 1 - 967 T + p^{3} T^{2} \)
71 \( 1 + 285 T + p^{3} T^{2} \)
73 \( 1 - 838 T + p^{3} T^{2} \)
79 \( 1 + 469 T + p^{3} T^{2} \)
83 \( 1 - 406 T + p^{3} T^{2} \)
89 \( 1 - 324 T + p^{3} T^{2} \)
97 \( 1 - 114 T + p^{3} T^{2} \)
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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

−10.47962296542795952564576634696, −9.703699336488039132238434977987, −8.658707706382875618606886022389, −7.70311081632331248256716098195, −6.56232493399649424470273919119, −5.48155291614182651935287692198, −4.92085966244524190500357276138, −3.91398616866517065064944274955, −2.59610125188529871055992797638, −0.72887306247798585303484237221, 0.72887306247798585303484237221, 2.59610125188529871055992797638, 3.91398616866517065064944274955, 4.92085966244524190500357276138, 5.48155291614182651935287692198, 6.56232493399649424470273919119, 7.70311081632331248256716098195, 8.658707706382875618606886022389, 9.703699336488039132238434977987, 10.47962296542795952564576634696

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