Properties

Label 2-150-1.1-c9-0-23
Degree $2$
Conductor $150$
Sign $-1$
Analytic cond. $77.2553$
Root an. cond. $8.78950$
Motivic weight $9$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 16·2-s − 81·3-s + 256·4-s − 1.29e3·6-s + 3.26e3·7-s + 4.09e3·8-s + 6.56e3·9-s − 7.04e4·11-s − 2.07e4·12-s + 6.17e4·13-s + 5.23e4·14-s + 6.55e4·16-s + 9.83e3·17-s + 1.04e5·18-s − 3.09e5·19-s − 2.64e5·21-s − 1.12e6·22-s − 4.01e5·23-s − 3.31e5·24-s + 9.88e5·26-s − 5.31e5·27-s + 8.36e5·28-s + 4.44e6·29-s − 8.99e5·31-s + 1.04e6·32-s + 5.70e6·33-s + 1.57e5·34-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.514·7-s + 0.353·8-s + 1/3·9-s − 1.45·11-s − 0.288·12-s + 0.600·13-s + 0.363·14-s + 1/4·16-s + 0.0285·17-s + 0.235·18-s − 0.545·19-s − 0.297·21-s − 1.02·22-s − 0.298·23-s − 0.204·24-s + 0.424·26-s − 0.192·27-s + 0.257·28-s + 1.16·29-s − 0.174·31-s + 0.176·32-s + 0.837·33-s + 0.0201·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(150\)    =    \(2 \cdot 3 \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(77.2553\)
Root analytic conductor: \(8.78950\)
Motivic weight: \(9\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 150,\ (\ :9/2),\ -1)\)

Particular Values

\(L(5)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - p^{4} T \)
3 \( 1 + p^{4} T \)
5 \( 1 \)
good7 \( 1 - 467 p T + p^{9} T^{2} \)
11 \( 1 + 70470 T + p^{9} T^{2} \)
13 \( 1 - 61793 T + p^{9} T^{2} \)
17 \( 1 - 9834 T + p^{9} T^{2} \)
19 \( 1 + 309709 T + p^{9} T^{2} \)
23 \( 1 + 401250 T + p^{9} T^{2} \)
29 \( 1 - 4442754 T + p^{9} T^{2} \)
31 \( 1 + 899467 T + p^{9} T^{2} \)
37 \( 1 + 15099982 T + p^{9} T^{2} \)
41 \( 1 - 15142944 T + p^{9} T^{2} \)
43 \( 1 + 357511 T + p^{9} T^{2} \)
47 \( 1 + 31748190 T + p^{9} T^{2} \)
53 \( 1 - 42870276 T + p^{9} T^{2} \)
59 \( 1 + 124820970 T + p^{9} T^{2} \)
61 \( 1 + 38947945 T + p^{9} T^{2} \)
67 \( 1 + 105967771 T + p^{9} T^{2} \)
71 \( 1 + 62841360 T + p^{9} T^{2} \)
73 \( 1 + 466543582 T + p^{9} T^{2} \)
79 \( 1 + 551941792 T + p^{9} T^{2} \)
83 \( 1 + 444962202 T + p^{9} T^{2} \)
89 \( 1 - 642049248 T + p^{9} T^{2} \)
97 \( 1 + 204187087 T + p^{9} T^{2} \)
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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

−10.86681824139689445605375725874, −10.23507265555584627077591491377, −8.516851617041301216776945551632, −7.49775038108804364270254218464, −6.25238335641351792003380369556, −5.28372589881167003928839117637, −4.37895979957852659313418506228, −2.90557704343679063268984474798, −1.56434196504651971868911146874, 0, 1.56434196504651971868911146874, 2.90557704343679063268984474798, 4.37895979957852659313418506228, 5.28372589881167003928839117637, 6.25238335641351792003380369556, 7.49775038108804364270254218464, 8.516851617041301216776945551632, 10.23507265555584627077591491377, 10.86681824139689445605375725874

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