Properties

Degree $2$
Conductor $33810$
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 + 8-s + 9-s − 10-s + 4·11-s − 12-s + 2·13-s + 15-s + 16-s + 6·17-s + 18-s − 4·19-s − 20-s + 4·22-s − 23-s − 24-s + 25-s + 2·26-s − 27-s − 2·29-s + 30-s + 32-s − 4·33-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.353·8-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 + 1.45·17-s + 0.235·18-s − 0.917·19-s − 0.223·20-s + 0.852·22-s − 0.208·23-s − 0.204·24-s + 1/5·25-s + 0.392·26-s − 0.192·27-s − 0.371·29-s + 0.182·30-s + 0.176·32-s − 0.696·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33810\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2} \cdot 23\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{33810} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 33810,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + T \)
5 \( 1 + T \)
7 \( 1 \)
23 \( 1 + T \)
good11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.11186905412656, −14.68000253152775, −14.37752623273320, −13.54362470863130, −13.21599125932684, −12.49243689205235, −12.03270528193227, −11.68058702632786, −11.28630862638616, −10.40637746815752, −10.23538278457385, −9.418037373306785, −8.673939106926573, −8.232674897741757, −7.475046940209126, −6.828287709434425, −6.523866624890095, −5.686927301888660, −5.426937148437428, −4.506492525963311, −3.984024279690777, −3.555464929495176, −2.794183918809160, −1.654727113557649, −1.215791178641572, 0, 1.215791178641572, 1.654727113557649, 2.794183918809160, 3.555464929495176, 3.984024279690777, 4.506492525963311, 5.426937148437428, 5.686927301888660, 6.523866624890095, 6.828287709434425, 7.475046940209126, 8.232674897741757, 8.673939106926573, 9.418037373306785, 10.23538278457385, 10.40637746815752, 11.28630862638616, 11.68058702632786, 12.03270528193227, 12.49243689205235, 13.21599125932684, 13.54362470863130, 14.37752623273320, 14.68000253152775, 15.11186905412656

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