Properties

Degree $2$
Conductor $205350$
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 + 8-s + 9-s + 4·11-s + 12-s − 2·13-s + 16-s + 2·17-s + 18-s − 4·19-s + 4·22-s − 8·23-s + 24-s − 2·26-s + 27-s + 2·29-s − 8·31-s + 32-s + 4·33-s + 2·34-s + 36-s − 4·38-s − 2·39-s + 10·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 1.20·11-s + 0.288·12-s − 0.554·13-s + 1/4·16-s + 0.485·17-s + 0.235·18-s − 0.917·19-s + 0.852·22-s − 1.66·23-s + 0.204·24-s − 0.392·26-s + 0.192·27-s + 0.371·29-s − 1.43·31-s + 0.176·32-s + 0.696·33-s + 0.342·34-s + 1/6·36-s − 0.648·38-s − 0.320·39-s + 1.56·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 205350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 205350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(205350\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 37^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{205350} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 205350,\ (\ :1/2),\ -1)\)

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 \( 1 \)
37 \( 1 \)
good7 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
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 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−13.27533479116419, −12.66744380065387, −12.42160610731977, −12.12859654845625, −11.45257382679314, −10.91180947439771, −10.66235820974307, −9.846132405816032, −9.397522719285198, −9.294698964943261, −8.316864393936228, −8.114107625785781, −7.492383576972960, −7.053874748358438, −6.384123064643504, −6.149770986505692, −5.486101081300483, −4.932265195477711, −4.166561924001206, −3.965768544484417, −3.564119851110602, −2.646507919952293, −2.309617923648356, −1.675031135242247, −1.027532356851654, 0, 1.027532356851654, 1.675031135242247, 2.309617923648356, 2.646507919952293, 3.564119851110602, 3.965768544484417, 4.166561924001206, 4.932265195477711, 5.486101081300483, 6.149770986505692, 6.384123064643504, 7.053874748358438, 7.492383576972960, 8.114107625785781, 8.316864393936228, 9.294698964943261, 9.397522719285198, 9.846132405816032, 10.66235820974307, 10.91180947439771, 11.45257382679314, 12.12859654845625, 12.42160610731977, 12.66744380065387, 13.27533479116419

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