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  − 2·2-s − 3-s + 2·4-s + 5-s + 2·6-s − 7-s + 9-s − 2·10-s − 3·11-s − 2·12-s − 6·13-s + 2·14-s − 15-s − 4·16-s − 2·18-s − 19-s + 2·20-s + 21-s + 6·22-s + 25-s + 12·26-s − 27-s − 2·28-s + 5·29-s + 2·30-s + 8·31-s + 8·32-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.577·3-s + 4-s + 0.447·5-s + 0.816·6-s − 0.377·7-s + 1/3·9-s − 0.632·10-s − 0.904·11-s − 0.577·12-s − 1.66·13-s + 0.534·14-s − 0.258·15-s − 16-s − 0.471·18-s − 0.229·19-s + 0.447·20-s + 0.218·21-s + 1.27·22-s + 1/5·25-s + 2.35·26-s − 0.192·27-s − 0.377·28-s + 0.928·29-s + 0.365·30-s + 1.43·31-s + 1.41·32-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 \( 1 + p T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 - 11 T + p T^{2} \)
67 \( 1 + 6 T + p T^{2} \)
71 \( 1 + 15 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
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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

−15.82811389524017, −15.02720785053160, −14.37956293949175, −13.76625582416581, −13.14711888908856, −12.70801935519858, −11.98521499075363, −11.63827345037075, −10.84489705090803, −10.23778783520398, −10.03158476905236, −9.724236041040383, −8.838937694199849, −8.455273480926633, −7.709924738173199, −7.263305664197827, −6.729194407768085, −6.133075114238305, −5.295675232669858, −4.826206619671803, −4.207943298934284, −2.872086778937287, −2.502711506421783, −1.664564381407011, −0.7185349995431105, 0, 0.7185349995431105, 1.664564381407011, 2.502711506421783, 2.872086778937287, 4.207943298934284, 4.826206619671803, 5.295675232669858, 6.133075114238305, 6.729194407768085, 7.263305664197827, 7.709924738173199, 8.455273480926633, 8.838937694199849, 9.724236041040383, 10.03158476905236, 10.23778783520398, 10.84489705090803, 11.63827345037075, 11.98521499075363, 12.70801935519858, 13.14711888908856, 13.76625582416581, 14.37956293949175, 15.02720785053160, 15.82811389524017

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