Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5^{2} \cdot 13^{2} $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 6-s − 4·7-s − 8-s + 9-s − 12-s + 4·14-s + 16-s − 6·17-s − 18-s + 4·19-s + 4·21-s + 24-s − 27-s − 4·28-s − 6·29-s − 8·31-s − 32-s + 6·34-s + 36-s + 2·37-s − 4·38-s + 6·41-s − 4·42-s + 4·43-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.288·12-s + 1.06·14-s + 1/4·16-s − 1.45·17-s − 0.235·18-s + 0.917·19-s + 0.872·21-s + 0.204·24-s − 0.192·27-s − 0.755·28-s − 1.11·29-s − 1.43·31-s − 0.176·32-s + 1.02·34-s + 1/6·36-s + 0.328·37-s − 0.648·38-s + 0.937·41-s − 0.617·42-s + 0.609·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(25350\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{25350} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 25350,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;13\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;13\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + T \)
5 \( 1 \)
13 \( 1 \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
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

−15.63419341575444, −15.46174667174345, −14.70803041368595, −13.90867786210219, −13.32383483435033, −12.79844027680818, −12.50429655192188, −11.68413136556548, −11.21081290834853, −10.72728087251998, −10.14902133063942, −9.469004409465197, −9.206283794352783, −8.716489344672148, −7.553974126520551, −7.374311025809978, −6.667574693436817, −6.099071546021984, −5.699308357019428, −4.834491586603662, −3.956941867760258, −3.399544233618051, −2.574731821651473, −1.849156494062333, −0.7286167505725691, 0, 0.7286167505725691, 1.849156494062333, 2.574731821651473, 3.399544233618051, 3.956941867760258, 4.834491586603662, 5.699308357019428, 6.099071546021984, 6.667574693436817, 7.374311025809978, 7.553974126520551, 8.716489344672148, 9.206283794352783, 9.469004409465197, 10.14902133063942, 10.72728087251998, 11.21081290834853, 11.68413136556548, 12.50429655192188, 12.79844027680818, 13.32383483435033, 13.90867786210219, 14.70803041368595, 15.46174667174345, 15.63419341575444

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