Properties

Label 2-2450-1.1-c1-0-55
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 8-s − 3·9-s − 4·11-s − 4.24·13-s + 16-s + 4.24·17-s − 3·18-s + 5.65·19-s − 4·22-s − 4.24·26-s − 4·29-s − 5.65·31-s + 32-s + 4.24·34-s − 3·36-s − 6·37-s + 5.65·38-s − 1.41·41-s − 12·43-s − 4·44-s − 4.24·52-s − 12·53-s − 4·58-s − 11.3·59-s + 7.07·61-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 0.353·8-s − 9-s − 1.20·11-s − 1.17·13-s + 0.250·16-s + 1.02·17-s − 0.707·18-s + 1.29·19-s − 0.852·22-s − 0.832·26-s − 0.742·29-s − 1.01·31-s + 0.176·32-s + 0.727·34-s − 0.5·36-s − 0.986·37-s + 0.917·38-s − 0.220·41-s − 1.82·43-s − 0.603·44-s − 0.588·52-s − 1.64·53-s − 0.525·58-s − 1.47·59-s + 0.905·61-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: $\chi_{2450} (1, \cdot )$
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 \( 1 + 3T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 + 4.24T + 13T^{2} \)
17 \( 1 - 4.24T + 17T^{2} \)
19 \( 1 - 5.65T + 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 4T + 29T^{2} \)
31 \( 1 + 5.65T + 31T^{2} \)
37 \( 1 + 6T + 37T^{2} \)
41 \( 1 + 1.41T + 41T^{2} \)
43 \( 1 + 12T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 12T + 53T^{2} \)
59 \( 1 + 11.3T + 59T^{2} \)
61 \( 1 - 7.07T + 61T^{2} \)
67 \( 1 + 12T + 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 - 4.24T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 16.9T + 83T^{2} \)
89 \( 1 + 4.24T + 89T^{2} \)
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

−8.363890546288383534446644373507, −7.64676227678662275011660042611, −7.13864555412994918477703332759, −5.93237179282087313858039130507, −5.28414901579079646322877809444, −4.89804885205522541521761378318, −3.39007701798454149535875122819, −2.97785655923287429873323717644, −1.84006062092222343779552731877, 0, 1.84006062092222343779552731877, 2.97785655923287429873323717644, 3.39007701798454149535875122819, 4.89804885205522541521761378318, 5.28414901579079646322877809444, 5.93237179282087313858039130507, 7.13864555412994918477703332759, 7.64676227678662275011660042611, 8.363890546288383534446644373507

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