Properties

Label 2-2450-1.1-c1-0-43
Degree $2$
Conductor $2450$
Sign $-1$
Analytic cond. $19.5633$
Root an. cond. $4.42304$
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 + 4-s − 8-s − 3·9-s + 3·11-s − 5·13-s + 16-s − 2·17-s + 3·18-s + 5·19-s − 3·22-s + 7·23-s + 5·26-s − 4·29-s + 2·31-s − 32-s + 2·34-s − 3·36-s − 37-s − 5·38-s − 3·41-s − 2·43-s + 3·44-s − 7·46-s − 7·47-s − 5·52-s − 9·53-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.353·8-s − 9-s + 0.904·11-s − 1.38·13-s + 1/4·16-s − 0.485·17-s + 0.707·18-s + 1.14·19-s − 0.639·22-s + 1.45·23-s + 0.980·26-s − 0.742·29-s + 0.359·31-s − 0.176·32-s + 0.342·34-s − 1/2·36-s − 0.164·37-s − 0.811·38-s − 0.468·41-s − 0.304·43-s + 0.452·44-s − 1.03·46-s − 1.02·47-s − 0.693·52-s − 1.23·53-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2450\)    =    \(2 \cdot 5^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(19.5633\)
Root analytic conductor: \(4.42304\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2450} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2450,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 - 7 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 + 12 T + p T^{2} \)
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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

−8.722288200303847181464711533795, −7.80471099931965682898184894578, −7.12191173900728537243214164656, −6.43773390159168975843176858276, −5.43945201706455509668151309560, −4.71835791677761993355466214212, −3.36202621499620939854951938574, −2.65167937662424348651501651075, −1.42353746843223523996833758028, 0, 1.42353746843223523996833758028, 2.65167937662424348651501651075, 3.36202621499620939854951938574, 4.71835791677761993355466214212, 5.43945201706455509668151309560, 6.43773390159168975843176858276, 7.12191173900728537243214164656, 7.80471099931965682898184894578, 8.722288200303847181464711533795

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