Properties

Label 2-675-1.1-c3-0-14
Degree $2$
Conductor $675$
Sign $1$
Analytic cond. $39.8262$
Root an. cond. $6.31080$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 4·4-s + 24·8-s − 10·11-s + 80·13-s − 16·16-s + 7·17-s − 113·19-s + 20·22-s − 81·23-s − 160·26-s + 220·29-s − 189·31-s − 160·32-s − 14·34-s − 170·37-s + 226·38-s + 130·41-s − 10·43-s + 40·44-s + 162·46-s + 160·47-s − 343·49-s − 320·52-s + 631·53-s − 440·58-s + 560·59-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 1.06·8-s − 0.274·11-s + 1.70·13-s − 1/4·16-s + 0.0998·17-s − 1.36·19-s + 0.193·22-s − 0.734·23-s − 1.20·26-s + 1.40·29-s − 1.09·31-s − 0.883·32-s − 0.0706·34-s − 0.755·37-s + 0.964·38-s + 0.495·41-s − 0.0354·43-s + 0.137·44-s + 0.519·46-s + 0.496·47-s − 49-s − 0.853·52-s + 1.63·53-s − 0.996·58-s + 1.23·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(39.8262\)
Root analytic conductor: \(6.31080\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{675} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.051549955\)
\(L(\frac12)\) \(\approx\) \(1.051549955\)
\(L(\frac{5}{2})\) 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 \( 1 \)
good2 \( 1 + p T + p^{3} T^{2} \)
7 \( 1 + p^{3} T^{2} \)
11 \( 1 + 10 T + p^{3} T^{2} \)
13 \( 1 - 80 T + p^{3} T^{2} \)
17 \( 1 - 7 T + p^{3} T^{2} \)
19 \( 1 + 113 T + p^{3} T^{2} \)
23 \( 1 + 81 T + p^{3} T^{2} \)
29 \( 1 - 220 T + p^{3} T^{2} \)
31 \( 1 + 189 T + p^{3} T^{2} \)
37 \( 1 + 170 T + p^{3} T^{2} \)
41 \( 1 - 130 T + p^{3} T^{2} \)
43 \( 1 + 10 T + p^{3} T^{2} \)
47 \( 1 - 160 T + p^{3} T^{2} \)
53 \( 1 - 631 T + p^{3} T^{2} \)
59 \( 1 - 560 T + p^{3} T^{2} \)
61 \( 1 - 229 T + p^{3} T^{2} \)
67 \( 1 + 750 T + p^{3} T^{2} \)
71 \( 1 + 890 T + p^{3} T^{2} \)
73 \( 1 - 890 T + p^{3} T^{2} \)
79 \( 1 + 27 T + p^{3} T^{2} \)
83 \( 1 - 429 T + p^{3} T^{2} \)
89 \( 1 - 750 T + p^{3} T^{2} \)
97 \( 1 - 1480 T + p^{3} T^{2} \)
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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

−10.25568621395669623652478502475, −8.959537146632473059915544997972, −8.588775636096517273164333831876, −7.77538129577762629732119764149, −6.61395261131403054882410943788, −5.66590465396959649829300645676, −4.45598214351314180131669587651, −3.60485700822879633842008529539, −1.92556283558731529427559757737, −0.67967671919860592609924455428, 0.67967671919860592609924455428, 1.92556283558731529427559757737, 3.60485700822879633842008529539, 4.45598214351314180131669587651, 5.66590465396959649829300645676, 6.61395261131403054882410943788, 7.77538129577762629732119764149, 8.588775636096517273164333831876, 8.959537146632473059915544997972, 10.25568621395669623652478502475

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