Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 \cdot 17^{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  + 3-s − 2·4-s + 5-s − 7-s + 9-s − 6·11-s − 2·12-s + 5·13-s + 15-s + 4·16-s − 2·20-s − 21-s + 23-s + 25-s + 27-s + 2·28-s − 2·29-s − 31-s − 6·33-s − 35-s − 2·36-s − 3·37-s + 5·39-s − 7·41-s − 6·43-s + 12·44-s + 45-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 1.80·11-s − 0.577·12-s + 1.38·13-s + 0.258·15-s + 16-s − 0.447·20-s − 0.218·21-s + 0.208·23-s + 1/5·25-s + 0.192·27-s + 0.377·28-s − 0.371·29-s − 0.179·31-s − 1.04·33-s − 0.169·35-s − 1/3·36-s − 0.493·37-s + 0.800·39-s − 1.09·41-s − 0.914·43-s + 1.80·44-s + 0.149·45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 30345 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30345 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(30345\)    =    \(3 \cdot 5 \cdot 7 \cdot 17^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{30345} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 30345,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7,\;17\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;5,\;7,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 + T \)
17 \( 1 \)
good2 \( 1 + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 + 7 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 - 8 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 + 13 T + p T^{2} \)
67 \( 1 + 6 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 - 15 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−15.23970170583884, −15.01585259592155, −14.07510548116685, −13.66393859003386, −13.34557011134988, −13.03333491767740, −12.45909410079717, −11.75022689169601, −10.80698053226329, −10.44157029082084, −10.08610958102203, −9.281417581532506, −8.959746987045670, −8.253432713533801, −8.037377497819323, −7.219831148630372, −6.542925507000620, −5.691303376584874, −5.371205575181298, −4.726973820035554, −3.866168459622225, −3.407106232721364, −2.723241385091451, −1.918252178392329, −0.9813202219127927, 0, 0.9813202219127927, 1.918252178392329, 2.723241385091451, 3.407106232721364, 3.866168459622225, 4.726973820035554, 5.371205575181298, 5.691303376584874, 6.542925507000620, 7.219831148630372, 8.037377497819323, 8.253432713533801, 8.959746987045670, 9.281417581532506, 10.08610958102203, 10.44157029082084, 10.80698053226329, 11.75022689169601, 12.45909410079717, 13.03333491767740, 13.34557011134988, 13.66393859003386, 14.07510548116685, 15.01585259592155, 15.23970170583884

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