Properties

Label 2-2450-1.1-c1-0-2
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 − 1.63·3-s + 4-s + 1.63·6-s − 8-s − 0.316·9-s − 1.31·11-s − 1.63·12-s − 6.10·13-s + 16-s − 2.60·17-s + 0.316·18-s − 3.05·19-s + 1.31·22-s − 4.63·23-s + 1.63·24-s + 6.10·26-s + 5.43·27-s + 10.6·29-s − 5.65·31-s − 32-s + 2.15·33-s + 2.60·34-s − 0.316·36-s − 8.63·37-s + 3.05·38-s + 10·39-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.945·3-s + 0.5·4-s + 0.668·6-s − 0.353·8-s − 0.105·9-s − 0.396·11-s − 0.472·12-s − 1.69·13-s + 0.250·16-s − 0.631·17-s + 0.0746·18-s − 0.700·19-s + 0.280·22-s − 0.966·23-s + 0.334·24-s + 1.19·26-s + 1.04·27-s + 1.97·29-s − 1.01·31-s − 0.176·32-s + 0.375·33-s + 0.446·34-s − 0.0527·36-s − 1.41·37-s + 0.495·38-s + 1.60·39-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

\(L(1)\) \(\approx\) \(0.3793835464\)
\(L(\frac12)\) \(\approx\) \(0.3793835464\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.63T + 3T^{2} \)
11 \( 1 + 1.31T + 11T^{2} \)
13 \( 1 + 6.10T + 13T^{2} \)
17 \( 1 + 2.60T + 17T^{2} \)
19 \( 1 + 3.05T + 19T^{2} \)
23 \( 1 + 4.63T + 23T^{2} \)
29 \( 1 - 10.6T + 29T^{2} \)
31 \( 1 + 5.65T + 31T^{2} \)
37 \( 1 + 8.63T + 37T^{2} \)
41 \( 1 - 7.29T + 41T^{2} \)
43 \( 1 - 6.63T + 43T^{2} \)
47 \( 1 - 3.27T + 47T^{2} \)
53 \( 1 + 8T + 53T^{2} \)
59 \( 1 + 7.51T + 59T^{2} \)
61 \( 1 + 3.27T + 61T^{2} \)
67 \( 1 - 11.6T + 67T^{2} \)
71 \( 1 + 2T + 71T^{2} \)
73 \( 1 + 2.60T + 73T^{2} \)
79 \( 1 - 14.6T + 79T^{2} \)
83 \( 1 + 12.9T + 83T^{2} \)
89 \( 1 + 9.15T + 89T^{2} \)
97 \( 1 - 4.24T + 97T^{2} \)
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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

−8.966033993254325128194453578446, −8.182057500019190910249892578418, −7.39520229610779740604221015725, −6.65713544732406862553084447638, −5.94656844142486364049030107273, −5.09075440200538523055893951284, −4.37069519026931108964095032958, −2.87210621576124059994467586693, −2.05367618060061561174683409001, −0.43026665370124702441174195180, 0.43026665370124702441174195180, 2.05367618060061561174683409001, 2.87210621576124059994467586693, 4.37069519026931108964095032958, 5.09075440200538523055893951284, 5.94656844142486364049030107273, 6.65713544732406862553084447638, 7.39520229610779740604221015725, 8.182057500019190910249892578418, 8.966033993254325128194453578446

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