Properties

Degree 2
Conductor $ 3 \cdot 5^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s − 4-s + 6-s − 3·8-s + 9-s − 4·11-s − 12-s + 2·13-s − 16-s − 2·17-s + 18-s + 4·19-s − 4·22-s − 3·24-s + 2·26-s + 27-s − 2·29-s + 5·32-s − 4·33-s − 2·34-s − 36-s + 10·37-s + 4·38-s + 2·39-s + 10·41-s − 4·43-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.408·6-s − 1.06·8-s + 1/3·9-s − 1.20·11-s − 0.288·12-s + 0.554·13-s − 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.917·19-s − 0.852·22-s − 0.612·24-s + 0.392·26-s + 0.192·27-s − 0.371·29-s + 0.883·32-s − 0.696·33-s − 0.342·34-s − 1/6·36-s + 1.64·37-s + 0.648·38-s + 0.320·39-s + 1.56·41-s − 0.609·43-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(75\)    =    \(3 \cdot 5^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{75} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 75,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.252737444$
$L(\frac12)$  $\approx$  $1.252737444$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;5\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;5\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 - T \)
5 \( 1 \)
good2 \( 1 - T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−19.63841597864066, −18.35321154397529, −17.98711144863008, −16.28749601122251, −15.33748864368926, −14.41099080497045, −13.37224542302975, −12.89741011334273, −11.42796116649064, −9.965958896202438, −8.843956595114681, −7.680038443857660, −5.864976538429658, −4.521604448848747, −3.015497841406864, 3.015497841406864, 4.521604448848747, 5.864976538429658, 7.680038443857660, 8.843956595114681, 9.965958896202438, 11.42796116649064, 12.89741011334273, 13.37224542302975, 14.41099080497045, 15.33748864368926, 16.28749601122251, 17.98711144863008, 18.35321154397529, 19.63841597864066

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