Properties

Label 2-2450-1.1-c3-0-40
Degree $2$
Conductor $2450$
Sign $1$
Analytic cond. $144.554$
Root an. cond. $12.0230$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 7·3-s + 4·4-s + 14·6-s − 8·8-s + 22·9-s − 37·11-s − 28·12-s + 51·13-s + 16·16-s + 41·17-s − 44·18-s + 108·19-s + 74·22-s + 70·23-s + 56·24-s − 102·26-s + 35·27-s − 249·29-s + 134·31-s − 32·32-s + 259·33-s − 82·34-s + 88·36-s + 334·37-s − 216·38-s − 357·39-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.34·3-s + 1/2·4-s + 0.952·6-s − 0.353·8-s + 0.814·9-s − 1.01·11-s − 0.673·12-s + 1.08·13-s + 1/4·16-s + 0.584·17-s − 0.576·18-s + 1.30·19-s + 0.717·22-s + 0.634·23-s + 0.476·24-s − 0.769·26-s + 0.249·27-s − 1.59·29-s + 0.776·31-s − 0.176·32-s + 1.36·33-s − 0.413·34-s + 0.407·36-s + 1.48·37-s − 0.922·38-s − 1.46·39-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: yes
Arithmetic: yes
Character: $\chi_{2450} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2450,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9003516372\)
\(L(\frac12)\) \(\approx\) \(0.9003516372\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + p T \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + 7 T + p^{3} T^{2} \)
11 \( 1 + 37 T + p^{3} T^{2} \)
13 \( 1 - 51 T + p^{3} T^{2} \)
17 \( 1 - 41 T + p^{3} T^{2} \)
19 \( 1 - 108 T + p^{3} T^{2} \)
23 \( 1 - 70 T + p^{3} T^{2} \)
29 \( 1 + 249 T + p^{3} T^{2} \)
31 \( 1 - 134 T + p^{3} T^{2} \)
37 \( 1 - 334 T + p^{3} T^{2} \)
41 \( 1 + 206 T + p^{3} T^{2} \)
43 \( 1 - 376 T + p^{3} T^{2} \)
47 \( 1 + 287 T + p^{3} T^{2} \)
53 \( 1 - 6 T + p^{3} T^{2} \)
59 \( 1 - 2 T + p^{3} T^{2} \)
61 \( 1 - 940 T + p^{3} T^{2} \)
67 \( 1 + 106 T + p^{3} T^{2} \)
71 \( 1 - 456 T + p^{3} T^{2} \)
73 \( 1 - 650 T + p^{3} T^{2} \)
79 \( 1 + 1239 T + p^{3} T^{2} \)
83 \( 1 - 428 T + p^{3} T^{2} \)
89 \( 1 - 220 T + p^{3} T^{2} \)
97 \( 1 + 1055 T + p^{3} T^{2} \)
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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

−8.527259785355391865168641107197, −7.78059352138516444045297451606, −7.09714703935566622624042169251, −6.17827302025823209895203088113, −5.58797735565882411243560914945, −4.99505154327013472725489095979, −3.70246339185228506312641729372, −2.67398808471436213284214440920, −1.27665075767756063101538773976, −0.57345904786561528599893467382, 0.57345904786561528599893467382, 1.27665075767756063101538773976, 2.67398808471436213284214440920, 3.70246339185228506312641729372, 4.99505154327013472725489095979, 5.58797735565882411243560914945, 6.17827302025823209895203088113, 7.09714703935566622624042169251, 7.78059352138516444045297451606, 8.527259785355391865168641107197

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