Properties

Label 2-127050-1.1-c1-0-148
Degree $2$
Conductor $127050$
Sign $-1$
Analytic cond. $1014.49$
Root an. cond. $31.8512$
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 − 12-s − 13-s − 14-s + 16-s + 3·17-s − 18-s + 4·19-s − 21-s + 3·23-s + 24-s + 26-s − 27-s + 28-s − 3·29-s + 5·31-s − 32-s − 3·34-s + 36-s + 10·37-s − 4·38-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 − 0.288·12-s − 0.277·13-s − 0.267·14-s + 1/4·16-s + 0.727·17-s − 0.235·18-s + 0.917·19-s − 0.218·21-s + 0.625·23-s + 0.204·24-s + 0.196·26-s − 0.192·27-s + 0.188·28-s − 0.557·29-s + 0.898·31-s − 0.176·32-s − 0.514·34-s + 1/6·36-s + 1.64·37-s − 0.648·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 127050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 127050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(127050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(1014.49\)
Root analytic conductor: \(31.8512\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{127050} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 127050,\ (\ :1/2),\ -1)\)

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 \( 1 \)
good13 \( 1 + T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 + 11 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−13.54159996375888, −13.37323795279270, −12.61210722148767, −12.14984061922194, −11.72880138539545, −11.33869641608543, −10.89120414044510, −10.27316725927916, −9.918428131867592, −9.417650118811560, −9.001342980824080, −8.173639858836437, −7.958908118925214, −7.328795244889025, −6.954802883559358, −6.275438501796151, −5.817496849546772, −5.229628224936555, −4.769173099571831, −4.152215756158090, −3.282018451597861, −2.910410273420802, −2.060316353769202, −1.348195243952484, −0.8900733792881443, 0, 0.8900733792881443, 1.348195243952484, 2.060316353769202, 2.910410273420802, 3.282018451597861, 4.152215756158090, 4.769173099571831, 5.229628224936555, 5.817496849546772, 6.275438501796151, 6.954802883559358, 7.328795244889025, 7.958908118925214, 8.173639858836437, 9.001342980824080, 9.417650118811560, 9.918428131867592, 10.27316725927916, 10.89120414044510, 11.33869641608543, 11.72880138539545, 12.14984061922194, 12.61210722148767, 13.37323795279270, 13.54159996375888

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