Properties

Degree 2
Conductor $ 2 \cdot 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-s − 3-s + 4-s − 5-s + 6-s − 7-s − 8-s + 9-s + 10-s − 12-s + 2·13-s + 14-s + 15-s + 16-s − 18-s + 8·19-s − 20-s + 21-s + 24-s + 25-s − 2·26-s − 27-s − 28-s − 6·29-s − 30-s + 4·31-s − 32-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s + 0.554·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s − 0.235·18-s + 1.83·19-s − 0.223·20-s + 0.218·21-s + 0.204·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s − 0.188·28-s − 1.11·29-s − 0.182·30-s + 0.718·31-s − 0.176·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(60690\)    =    \(2 \cdot 3 \cdot 5 \cdot 7 \cdot 17^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{60690} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 60690,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7,\;17\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + T \)
5 \( 1 + T \)
7 \( 1 + T \)
17 \( 1 \)
good11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
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 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−14.71386165115577, −14.05664862065829, −13.36094685426302, −13.08847742771532, −12.32077850031928, −11.92087565133911, −11.41938365414746, −11.05482681031517, −10.54722627330697, −9.798119738467615, −9.516896455073921, −9.054504993793463, −8.243853308154316, −7.771882358848491, −7.363212396783897, −6.787497948518891, −6.070327513415552, −5.797541351329339, −5.011672850567599, −4.392604693886512, −3.614634780352282, −3.131760580252791, −2.389800273438028, −1.368369037127241, −0.8944848116094486, 0, 0.8944848116094486, 1.368369037127241, 2.389800273438028, 3.131760580252791, 3.614634780352282, 4.392604693886512, 5.011672850567599, 5.797541351329339, 6.070327513415552, 6.787497948518891, 7.363212396783897, 7.771882358848491, 8.243853308154316, 9.054504993793463, 9.516896455073921, 9.798119738467615, 10.54722627330697, 11.05482681031517, 11.41938365414746, 11.92087565133911, 12.32077850031928, 13.08847742771532, 13.36094685426302, 14.05664862065829, 14.71386165115577

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