Properties

Label 2-11550-1.1-c1-0-33
Degree $2$
Conductor $11550$
Sign $-1$
Analytic cond. $92.2272$
Root an. cond. $9.60350$
Motivic weight $1$
Arithmetic yes
Rational yes
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 − 7-s − 8-s + 9-s + 11-s − 12-s + 2·13-s + 14-s + 16-s + 2·17-s − 18-s − 6·19-s + 21-s − 22-s + 24-s − 2·26-s − 27-s − 28-s + 8·31-s − 32-s − 33-s − 2·34-s + 36-s − 2·37-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.301·11-s − 0.288·12-s + 0.554·13-s + 0.267·14-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 1.37·19-s + 0.218·21-s − 0.213·22-s + 0.204·24-s − 0.392·26-s − 0.192·27-s − 0.188·28-s + 1.43·31-s − 0.176·32-s − 0.174·33-s − 0.342·34-s + 1/6·36-s − 0.328·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(11550\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11\)
Sign: $-1$
Analytic conductor: \(92.2272\)
Root analytic conductor: \(9.60350\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{11550} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 11550,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 + T \)
11 \( 1 - T \)
good13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−16.80357278273145, −16.18103595326756, −15.83669999963331, −15.07800780719029, −14.67778362757503, −13.77083109645647, −13.25108620966529, −12.54016858321571, −12.08313390037835, −11.42511589185533, −10.92396846026270, −10.22307481092352, −9.902270206968222, −9.126336719032790, −8.404256130409997, −8.072808405923134, −7.037676780045465, −6.585064345758375, −6.093887159600340, −5.325036702716607, −4.462733921498354, −3.719932555817605, −2.872324816158318, −1.896239649800356, −1.037792526351240, 0, 1.037792526351240, 1.896239649800356, 2.872324816158318, 3.719932555817605, 4.462733921498354, 5.325036702716607, 6.093887159600340, 6.585064345758375, 7.037676780045465, 8.072808405923134, 8.404256130409997, 9.126336719032790, 9.902270206968222, 10.22307481092352, 10.92396846026270, 11.42511589185533, 12.08313390037835, 12.54016858321571, 13.25108620966529, 13.77083109645647, 14.67778362757503, 15.07800780719029, 15.83669999963331, 16.18103595326756, 16.80357278273145

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