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Properties

Label 2-2450-5.4-c1-0-40
Degree $2$
Conductor $2450$
Sign $0.894 + 0.447i$
Analytic cond. $19.5633$
Root an. cond. $4.42304$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s + 3i·3-s − 4-s − 3·6-s − i·8-s − 6·9-s − 2·11-s − 3i·12-s + 16-s + 4i·17-s − 6i·18-s − 6·19-s − 2i·22-s − 3i·23-s + 3·24-s + ⋯
L(s)  = 1  + 0.707i·2-s + 1.73i·3-s − 0.5·4-s − 1.22·6-s − 0.353i·8-s − 2·9-s − 0.603·11-s − 0.866i·12-s + 0.250·16-s + 0.970i·17-s − 1.41i·18-s − 1.37·19-s − 0.426i·22-s − 0.625i·23-s + 0.612·24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2450\)    =    \(2 \cdot 5^{2} \cdot 7^{2}\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(19.5633\)
Root analytic conductor: \(4.42304\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2450} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 2450,\ (\ :1/2),\ 0.894 + 0.447i)\)

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 \( 1 - iT \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 - 3iT - 3T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 4iT - 17T^{2} \)
19 \( 1 + 6T + 19T^{2} \)
23 \( 1 + 3iT - 23T^{2} \)
29 \( 1 + 9T + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + 4iT - 37T^{2} \)
41 \( 1 - 7T + 41T^{2} \)
43 \( 1 - 5iT - 43T^{2} \)
47 \( 1 + 8iT - 47T^{2} \)
53 \( 1 - 2iT - 53T^{2} \)
59 \( 1 - 10T + 59T^{2} \)
61 \( 1 + T + 61T^{2} \)
67 \( 1 + 9iT - 67T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + 4iT - 73T^{2} \)
79 \( 1 + 10T + 79T^{2} \)
83 \( 1 + 7iT - 83T^{2} \)
89 \( 1 - T + 89T^{2} \)
97 \( 1 + 14iT - 97T^{2} \)
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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

−8.813836587196226226741355963366, −8.375993269772858270578799366691, −7.46695195640450020994060669933, −6.27128022298553874526164451131, −5.72069370648719354687904553888, −4.85335017839183274127865422443, −4.19629552622464142466973928975, −3.55541401028347523403780501389, −2.31871374831885217339367391228, 0, 1.16717194110638346518059984721, 2.21241915725933517271838017580, 2.76768342325529352061347305153, 4.00669417617934749328539377673, 5.20603240951320501985454212796, 5.94132964106281671931828193920, 6.82512007805533580885341957655, 7.51709438543367188847806807998, 8.141716446630987438498307722048

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