Properties

Label 2-2450-1.1-c1-0-18
Degree $2$
Conductor $2450$
Sign $1$
Analytic cond. $19.5633$
Root an. cond. $4.42304$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 2.64·3-s + 4-s − 2.64·6-s + 8-s + 4.00·9-s + 5·11-s − 2.64·12-s + 5.29·13-s + 16-s − 2.64·17-s + 4.00·18-s − 7.93·19-s + 5·22-s − 4·23-s − 2.64·24-s + 5.29·26-s − 2.64·27-s + 6·29-s + 10.5·31-s + 32-s − 13.2·33-s − 2.64·34-s + 4.00·36-s − 4·37-s − 7.93·38-s − 14.0·39-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.52·3-s + 0.5·4-s − 1.08·6-s + 0.353·8-s + 1.33·9-s + 1.50·11-s − 0.763·12-s + 1.46·13-s + 0.250·16-s − 0.641·17-s + 0.942·18-s − 1.82·19-s + 1.06·22-s − 0.834·23-s − 0.540·24-s + 1.03·26-s − 0.509·27-s + 1.11·29-s + 1.90·31-s + 0.176·32-s − 2.30·33-s − 0.453·34-s + 0.666·36-s − 0.657·37-s − 1.28·38-s − 2.24·39-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: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2450,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.841287261\)
\(L(\frac12)\) \(\approx\) \(1.841287261\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.64T + 3T^{2} \)
11 \( 1 - 5T + 11T^{2} \)
13 \( 1 - 5.29T + 13T^{2} \)
17 \( 1 + 2.64T + 17T^{2} \)
19 \( 1 + 7.93T + 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 - 10.5T + 31T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 - 2.64T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + 5.29T + 47T^{2} \)
53 \( 1 - 4T + 53T^{2} \)
59 \( 1 + 5.29T + 59T^{2} \)
61 \( 1 - 5.29T + 61T^{2} \)
67 \( 1 - 5T + 67T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 - 2.64T + 73T^{2} \)
79 \( 1 - 10T + 79T^{2} \)
83 \( 1 - 2.64T + 83T^{2} \)
89 \( 1 - 13.2T + 89T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−8.804317404253944380183838237818, −8.238453358643676074796464535368, −6.74331653574359212690083389660, −6.30494996008142518305942590146, −6.20378140078212591614595264169, −4.90123892865172734409730900354, −4.32142764870998178576054646880, −3.58508012431733577971752563188, −1.98028389622359504742818949135, −0.875391699215869102345774678715, 0.875391699215869102345774678715, 1.98028389622359504742818949135, 3.58508012431733577971752563188, 4.32142764870998178576054646880, 4.90123892865172734409730900354, 6.20378140078212591614595264169, 6.30494996008142518305942590146, 6.74331653574359212690083389660, 8.238453358643676074796464535368, 8.804317404253944380183838237818

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