Properties

Label 2-2450-1.1-c3-0-102
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 7.28·3-s + 4·4-s − 14.5·6-s − 8·8-s + 26.0·9-s + 41.1·11-s + 29.1·12-s + 24.5·13-s + 16·16-s + 93.3·17-s − 52.0·18-s + 54.6·19-s − 82.2·22-s + 136.·23-s − 58.2·24-s − 49.1·26-s − 7.11·27-s + 282.·29-s − 54.6·31-s − 32·32-s + 299.·33-s − 186.·34-s + 104.·36-s − 212.·37-s − 109.·38-s + 179.·39-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.40·3-s + 0.5·4-s − 0.990·6-s − 0.353·8-s + 0.963·9-s + 1.12·11-s + 0.700·12-s + 0.524·13-s + 0.250·16-s + 1.33·17-s − 0.681·18-s + 0.659·19-s − 0.796·22-s + 1.23·23-s − 0.495·24-s − 0.370·26-s − 0.0507·27-s + 1.81·29-s − 0.316·31-s − 0.176·32-s + 1.57·33-s − 0.942·34-s + 0.481·36-s − 0.943·37-s − 0.466·38-s + 0.735·39-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: \(0\)
Selberg data: \((2,\ 2450,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(3.686668929\)
\(L(\frac12)\) \(\approx\) \(3.686668929\)
\(L(\frac{5}{2})\) 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 \( 1 - 7.28T + 27T^{2} \)
11 \( 1 - 41.1T + 1.33e3T^{2} \)
13 \( 1 - 24.5T + 2.19e3T^{2} \)
17 \( 1 - 93.3T + 4.91e3T^{2} \)
19 \( 1 - 54.6T + 6.85e3T^{2} \)
23 \( 1 - 136.T + 1.21e4T^{2} \)
29 \( 1 - 282.T + 2.43e4T^{2} \)
31 \( 1 + 54.6T + 2.97e4T^{2} \)
37 \( 1 + 212.T + 5.06e4T^{2} \)
41 \( 1 - 417.T + 6.89e4T^{2} \)
43 \( 1 + 193.T + 7.95e4T^{2} \)
47 \( 1 + 126.T + 1.03e5T^{2} \)
53 \( 1 + 437.T + 1.48e5T^{2} \)
59 \( 1 - 419.T + 2.05e5T^{2} \)
61 \( 1 + 323.T + 2.26e5T^{2} \)
67 \( 1 + 57.2T + 3.00e5T^{2} \)
71 \( 1 + 1.06e3T + 3.57e5T^{2} \)
73 \( 1 + 687.T + 3.89e5T^{2} \)
79 \( 1 - 716.T + 4.93e5T^{2} \)
83 \( 1 + 236.T + 5.71e5T^{2} \)
89 \( 1 + 457.T + 7.04e5T^{2} \)
97 \( 1 + 1.83e3T + 9.12e5T^{2} \)
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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

−8.678983416511753066570393841287, −8.011365211511479327541923580322, −7.31045600247057977011444573836, −6.58024822523047338828379985053, −5.59041875458284939187060121300, −4.36869180332694983051765130769, −3.30250858723989247657846670325, −2.95910438581192222113092554154, −1.62626585717505399475814276201, −0.973276918638944429877813337597, 0.973276918638944429877813337597, 1.62626585717505399475814276201, 2.95910438581192222113092554154, 3.30250858723989247657846670325, 4.36869180332694983051765130769, 5.59041875458284939187060121300, 6.58024822523047338828379985053, 7.31045600247057977011444573836, 8.011365211511479327541923580322, 8.678983416511753066570393841287

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