Properties

Label 2-2450-1.1-c3-0-115
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 − 3.07·3-s + 4·4-s + 6.15·6-s − 8·8-s − 17.5·9-s + 61.2·11-s − 12.3·12-s − 23.4·13-s + 16·16-s + 135.·17-s + 35.0·18-s − 62.0·19-s − 122.·22-s − 73.7·23-s + 24.6·24-s + 46.9·26-s + 137.·27-s − 303.·29-s − 172.·31-s − 32·32-s − 188.·33-s − 271.·34-s − 70.1·36-s + 381.·37-s + 124.·38-s + 72.2·39-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.592·3-s + 0.5·4-s + 0.418·6-s − 0.353·8-s − 0.649·9-s + 1.67·11-s − 0.296·12-s − 0.501·13-s + 0.250·16-s + 1.93·17-s + 0.459·18-s − 0.749·19-s − 1.18·22-s − 0.669·23-s + 0.209·24-s + 0.354·26-s + 0.976·27-s − 1.94·29-s − 0.998·31-s − 0.176·32-s − 0.994·33-s − 1.36·34-s − 0.324·36-s + 1.69·37-s + 0.530·38-s + 0.296·39-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 \( 1 + 3.07T + 27T^{2} \)
11 \( 1 - 61.2T + 1.33e3T^{2} \)
13 \( 1 + 23.4T + 2.19e3T^{2} \)
17 \( 1 - 135.T + 4.91e3T^{2} \)
19 \( 1 + 62.0T + 6.85e3T^{2} \)
23 \( 1 + 73.7T + 1.21e4T^{2} \)
29 \( 1 + 303.T + 2.43e4T^{2} \)
31 \( 1 + 172.T + 2.97e4T^{2} \)
37 \( 1 - 381.T + 5.06e4T^{2} \)
41 \( 1 + 163.T + 6.89e4T^{2} \)
43 \( 1 - 174.T + 7.95e4T^{2} \)
47 \( 1 + 225.T + 1.03e5T^{2} \)
53 \( 1 + 194.T + 1.48e5T^{2} \)
59 \( 1 + 72.0T + 2.05e5T^{2} \)
61 \( 1 - 163.T + 2.26e5T^{2} \)
67 \( 1 - 114.T + 3.00e5T^{2} \)
71 \( 1 - 442.T + 3.57e5T^{2} \)
73 \( 1 + 805.T + 3.89e5T^{2} \)
79 \( 1 - 259.T + 4.93e5T^{2} \)
83 \( 1 - 1.06e3T + 5.71e5T^{2} \)
89 \( 1 - 1.03e3T + 7.04e5T^{2} \)
97 \( 1 + 749.T + 9.12e5T^{2} \)
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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

−8.146189320050014999505774859639, −7.56501347780141130882247827072, −6.62234501094100561388267261916, −5.96696443774636890850896274124, −5.34594207719542584283484895514, −4.07070202885044698535156823295, −3.28500079657073747777274706457, −2.00458202521192126699497184688, −1.05445471980170585175473728205, 0, 1.05445471980170585175473728205, 2.00458202521192126699497184688, 3.28500079657073747777274706457, 4.07070202885044698535156823295, 5.34594207719542584283484895514, 5.96696443774636890850896274124, 6.62234501094100561388267261916, 7.56501347780141130882247827072, 8.146189320050014999505774859639

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