Properties

Label 2-150-1.1-c11-0-1
Degree $2$
Conductor $150$
Sign $1$
Analytic cond. $115.251$
Root an. cond. $10.7355$
Motivic weight $11$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 32·2-s − 243·3-s + 1.02e3·4-s + 7.77e3·6-s − 3.29e4·7-s − 3.27e4·8-s + 5.90e4·9-s − 7.58e5·11-s − 2.48e5·12-s + 2.48e6·13-s + 1.05e6·14-s + 1.04e6·16-s − 8.29e6·17-s − 1.88e6·18-s − 1.08e7·19-s + 8.00e6·21-s + 2.42e7·22-s − 2.05e7·23-s + 7.96e6·24-s − 7.94e7·26-s − 1.43e7·27-s − 3.37e7·28-s + 2.88e7·29-s + 1.50e8·31-s − 3.35e7·32-s + 1.84e8·33-s + 2.65e8·34-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.740·7-s − 0.353·8-s + 1/3·9-s − 1.42·11-s − 0.288·12-s + 1.85·13-s + 0.523·14-s + 1/4·16-s − 1.41·17-s − 0.235·18-s − 1.00·19-s + 0.427·21-s + 1.00·22-s − 0.665·23-s + 0.204·24-s − 1.31·26-s − 0.192·27-s − 0.370·28-s + 0.260·29-s + 0.944·31-s − 0.176·32-s + 0.820·33-s + 1.00·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(150\)    =    \(2 \cdot 3 \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(115.251\)
Root analytic conductor: \(10.7355\)
Motivic weight: \(11\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 150,\ (\ :11/2),\ 1)\)

Particular Values

\(L(6)\) \(\approx\) \(0.4422729502\)
\(L(\frac12)\) \(\approx\) \(0.4422729502\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + p^{5} T \)
3 \( 1 + p^{5} T \)
5 \( 1 \)
good7 \( 1 + 32936 T + p^{11} T^{2} \)
11 \( 1 + 758748 T + p^{11} T^{2} \)
13 \( 1 - 2482858 T + p^{11} T^{2} \)
17 \( 1 + 8290386 T + p^{11} T^{2} \)
19 \( 1 + 10867300 T + p^{11} T^{2} \)
23 \( 1 + 20539272 T + p^{11} T^{2} \)
29 \( 1 - 28814550 T + p^{11} T^{2} \)
31 \( 1 - 150501392 T + p^{11} T^{2} \)
37 \( 1 - 8645722 p T + p^{11} T^{2} \)
41 \( 1 + 368008998 T + p^{11} T^{2} \)
43 \( 1 + 620469572 T + p^{11} T^{2} \)
47 \( 1 + 2763110256 T + p^{11} T^{2} \)
53 \( 1 - 268284258 T + p^{11} T^{2} \)
59 \( 1 - 1672894740 T + p^{11} T^{2} \)
61 \( 1 + 7787197498 T + p^{11} T^{2} \)
67 \( 1 + 18706694156 T + p^{11} T^{2} \)
71 \( 1 + 8346990888 T + p^{11} T^{2} \)
73 \( 1 + 19641746522 T + p^{11} T^{2} \)
79 \( 1 + 5873807200 T + p^{11} T^{2} \)
83 \( 1 + 8492558172 T + p^{11} T^{2} \)
89 \( 1 - 75527864010 T + p^{11} T^{2} \)
97 \( 1 - 82356782494 T + p^{11} T^{2} \)
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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

−10.76005485732786193333903338172, −10.14232091097595963234289042221, −8.827076715795658522069799488524, −8.022536562809858036753374282923, −6.56607966442649288587107092926, −6.04220933144258190984858582892, −4.49869696222175789495463639439, −3.06318094364371104717337454373, −1.76182215492518012092477009204, −0.34795056068503844723200204445, 0.34795056068503844723200204445, 1.76182215492518012092477009204, 3.06318094364371104717337454373, 4.49869696222175789495463639439, 6.04220933144258190984858582892, 6.56607966442649288587107092926, 8.022536562809858036753374282923, 8.827076715795658522069799488524, 10.14232091097595963234289042221, 10.76005485732786193333903338172

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