Properties

Label 2-2450-1.1-c3-0-134
Degree $2$
Conductor $2450$
Sign $-1$
Analytic cond. $144.554$
Root an. cond. $12.0230$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 7.07·3-s + 4·4-s − 14.1·6-s + 8·8-s + 23.0·9-s − 14·11-s − 28.2·12-s + 50.9·13-s + 16·16-s + 1.41·17-s + 46.0·18-s + 1.41·19-s − 28·22-s − 140·23-s − 56.5·24-s + 101.·26-s + 28.2·27-s − 286·29-s + 93.3·31-s + 32·32-s + 98.9·33-s + 2.82·34-s + 92.0·36-s + 38·37-s + 2.82·38-s − 360·39-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.36·3-s + 0.5·4-s − 0.962·6-s + 0.353·8-s + 0.851·9-s − 0.383·11-s − 0.680·12-s + 1.08·13-s + 0.250·16-s + 0.0201·17-s + 0.602·18-s + 0.0170·19-s − 0.271·22-s − 1.26·23-s − 0.481·24-s + 0.768·26-s + 0.201·27-s − 1.83·29-s + 0.540·31-s + 0.176·32-s + 0.522·33-s + 0.0142·34-s + 0.425·36-s + 0.168·37-s + 0.0120·38-s − 1.47·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2450\)    =    \(2 \cdot 5^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(144.554\)
Root analytic conductor: \(12.0230\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2450,\ (\ :3/2),\ -1)\)

Particular Values

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

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2T \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + 7.07T + 27T^{2} \)
11 \( 1 + 14T + 1.33e3T^{2} \)
13 \( 1 - 50.9T + 2.19e3T^{2} \)
17 \( 1 - 1.41T + 4.91e3T^{2} \)
19 \( 1 - 1.41T + 6.85e3T^{2} \)
23 \( 1 + 140T + 1.21e4T^{2} \)
29 \( 1 + 286T + 2.43e4T^{2} \)
31 \( 1 - 93.3T + 2.97e4T^{2} \)
37 \( 1 - 38T + 5.06e4T^{2} \)
41 \( 1 - 125.T + 6.89e4T^{2} \)
43 \( 1 - 34T + 7.95e4T^{2} \)
47 \( 1 - 523.T + 1.03e5T^{2} \)
53 \( 1 - 74T + 1.48e5T^{2} \)
59 \( 1 + 434.T + 2.05e5T^{2} \)
61 \( 1 + 14.1T + 2.26e5T^{2} \)
67 \( 1 + 684T + 3.00e5T^{2} \)
71 \( 1 - 588T + 3.57e5T^{2} \)
73 \( 1 + 270.T + 3.89e5T^{2} \)
79 \( 1 - 1.22e3T + 4.93e5T^{2} \)
83 \( 1 - 422.T + 5.71e5T^{2} \)
89 \( 1 + 618.T + 7.04e5T^{2} \)
97 \( 1 - 1.48e3T + 9.12e5T^{2} \)
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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

−7.972573588764912874047333041868, −7.25251042351175463691371907292, −6.20545877091659162545689375240, −5.93622471832302949160875645892, −5.20725043161734134037000453510, −4.30091462727367112306503724482, −3.56984457303778247217196761589, −2.28411736587091172824019237619, −1.14037220585578311664133561774, 0, 1.14037220585578311664133561774, 2.28411736587091172824019237619, 3.56984457303778247217196761589, 4.30091462727367112306503724482, 5.20725043161734134037000453510, 5.93622471832302949160875645892, 6.20545877091659162545689375240, 7.25251042351175463691371907292, 7.972573588764912874047333041868

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