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Properties

Label 2-2450-5.4-c1-0-54
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 − i·3-s − 4-s + 6-s − i·8-s + 2·9-s − 6·11-s + i·12-s + 4i·13-s + 16-s + 2i·18-s + 2·19-s − 6i·22-s − 3i·23-s − 24-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 0.408·6-s − 0.353i·8-s + 0.666·9-s − 1.80·11-s + 0.288i·12-s + 1.10i·13-s + 0.250·16-s + 0.471i·18-s + 0.458·19-s − 1.27i·22-s − 0.625i·23-s − 0.204·24-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 \( 1 + iT - 3T^{2} \)
11 \( 1 + 6T + 11T^{2} \)
13 \( 1 - 4iT - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 + 3iT - 23T^{2} \)
29 \( 1 - 3T + 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 - 4iT - 37T^{2} \)
41 \( 1 + 9T + 41T^{2} \)
43 \( 1 + 7iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 + 6T + 59T^{2} \)
61 \( 1 + 5T + 61T^{2} \)
67 \( 1 + 5iT - 67T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 - 16iT - 73T^{2} \)
79 \( 1 + 2T + 79T^{2} \)
83 \( 1 + 3iT - 83T^{2} \)
89 \( 1 + 15T + 89T^{2} \)
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

−8.368101838091213956096238172155, −7.75970196401084028857125162275, −7.04999849863942220506086099363, −6.55231658877963995169320636457, −5.45897309365227582328132280412, −4.86394865906334225918541512063, −3.90386514363511901233331297291, −2.65676149105354880550586291193, −1.58967331109431555723055578923, 0, 1.53756854545559649779324046382, 2.80736117778798706238243893497, 3.40001472705529716918590440443, 4.49876568912956998969503122906, 5.21074142287028842400717180637, 5.77978045789144607023589471909, 7.26712398118375657056885622117, 7.77090568660566375346849901022, 8.596495746678319966689387101241

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