Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 53^{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 − 5-s + 6-s − 3·8-s + 9-s − 10-s − 4·11-s − 12-s − 2·13-s − 15-s − 16-s + 2·17-s + 18-s − 4·19-s + 20-s − 4·22-s − 3·24-s + 25-s − 2·26-s + 27-s − 2·29-s − 30-s + 5·32-s − 4·33-s + 2·34-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.447·5-s + 0.408·6-s − 1.06·8-s + 1/3·9-s − 0.316·10-s − 1.20·11-s − 0.288·12-s − 0.554·13-s − 0.258·15-s − 1/4·16-s + 0.485·17-s + 0.235·18-s − 0.917·19-s + 0.223·20-s − 0.852·22-s − 0.612·24-s + 1/5·25-s − 0.392·26-s + 0.192·27-s − 0.371·29-s − 0.182·30-s + 0.883·32-s − 0.696·33-s + 0.342·34-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 42135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 42135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(42135\)    =    \(3 \cdot 5 \cdot 53^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{42135} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 42135,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.7695504950$
$L(\frac12)$  $\approx$  $0.7695504950$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;5,\;53\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;5,\;53\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 - T \)
5 \( 1 + T \)
53 \( 1 \)
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 \( 1 - 8 T + p T^{2} \)
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

−14.71257491131218, −14.18017117423733, −13.69421244600120, −13.17572495169030, −12.79904255028319, −12.20902605321420, −11.95139154021268, −11.05444854267630, −10.44935400897144, −10.07055134110392, −9.408398574975771, −8.767012574800299, −8.392917483718792, −7.840647391999265, −7.250449647498366, −6.670995190534341, −5.798778119464859, −5.333539899629213, −4.750802841737688, −4.243545203846949, −3.550713454797098, −3.036619319656043, −2.440164482177408, −1.563657628307371, −0.2629916749897071, 0.2629916749897071, 1.563657628307371, 2.440164482177408, 3.036619319656043, 3.550713454797098, 4.243545203846949, 4.750802841737688, 5.333539899629213, 5.798778119464859, 6.670995190534341, 7.250449647498366, 7.840647391999265, 8.392917483718792, 8.767012574800299, 9.408398574975771, 10.07055134110392, 10.44935400897144, 11.05444854267630, 11.95139154021268, 12.20902605321420, 12.79904255028319, 13.17572495169030, 13.69421244600120, 14.18017117423733, 14.71257491131218

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