Properties

Label 2-75150-1.1-c1-0-43
Degree $2$
Conductor $75150$
Sign $1$
Analytic cond. $600.075$
Root an. cond. $24.4964$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 2·7-s − 8-s − 2·13-s + 2·14-s + 16-s − 6·17-s + 4·19-s − 8·23-s + 2·26-s − 2·28-s + 4·31-s − 32-s + 6·34-s − 10·37-s − 4·38-s − 4·41-s + 2·43-s + 8·46-s − 8·47-s − 3·49-s − 2·52-s + 6·53-s + 2·56-s − 10·59-s − 6·61-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.755·7-s − 0.353·8-s − 0.554·13-s + 0.534·14-s + 1/4·16-s − 1.45·17-s + 0.917·19-s − 1.66·23-s + 0.392·26-s − 0.377·28-s + 0.718·31-s − 0.176·32-s + 1.02·34-s − 1.64·37-s − 0.648·38-s − 0.624·41-s + 0.304·43-s + 1.17·46-s − 1.16·47-s − 3/7·49-s − 0.277·52-s + 0.824·53-s + 0.267·56-s − 1.30·59-s − 0.768·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 167\)
Sign: $1$
Analytic conductor: \(600.075\)
Root analytic conductor: \(24.4964\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 75150,\ (\ :1/2),\ 1)\)

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 \( 1 \)
5 \( 1 \)
167 \( 1 + T \)
good7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 6 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
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

−14.68625460600263, −13.81138486640133, −13.71357579044319, −13.11451796661131, −12.36355150222268, −11.99836862960482, −11.69829231081743, −10.88505430302078, −10.47530509010954, −9.983421945001246, −9.517231186699245, −9.076678259283749, −8.538289559959381, −7.911314609483100, −7.507516359585561, −6.753398217691754, −6.494454255480066, −5.925369806707540, −5.156741495881430, −4.620380489305179, −3.842102177675647, −3.269844250264857, −2.601532947661376, −1.987936228203936, −1.307225012402714, 0, 0, 1.307225012402714, 1.987936228203936, 2.601532947661376, 3.269844250264857, 3.842102177675647, 4.620380489305179, 5.156741495881430, 5.925369806707540, 6.494454255480066, 6.753398217691754, 7.507516359585561, 7.911314609483100, 8.538289559959381, 9.076678259283749, 9.517231186699245, 9.983421945001246, 10.47530509010954, 10.88505430302078, 11.69829231081743, 11.99836862960482, 12.36355150222268, 13.11451796661131, 13.71357579044319, 13.81138486640133, 14.68625460600263

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