Properties

Label 2-2450-5.4-c1-0-60
Degree $2$
Conductor $2450$
Sign $-0.447 - 0.894i$
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 − 5·11-s + 3i·12-s − 6i·13-s + 16-s + i·17-s − 6i·18-s − 3·19-s − 5i·22-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 − 1.50·11-s + 0.866i·12-s − 1.66i·13-s + 0.250·16-s + 0.242i·17-s − 1.41i·18-s − 0.688·19-s − 1.06i·22-s − 0.612·24-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2450\)    =    \(2 \cdot 5^{2} \cdot 7^{2}\)
Sign: $-0.447 - 0.894i$
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: \((2,\ 2450,\ (\ :1/2),\ -0.447 - 0.894i)\)

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

−8.240062220808379565180363808354, −7.75221844997496956988729162755, −6.88813602571735589824037104794, −6.27776705825829606733848795820, −5.51659758232396871381930395763, −4.83535015245104865470929570050, −3.15619455694130266348025053715, −2.49071683744559344059936404983, −1.11675087360072396431518897735, 0, 2.12181144244860667873958093578, 2.97191928692346512057964156740, 3.91841885518190209386700625795, 4.60487297833844137013602898135, 5.11002405248045217891381302415, 6.07165830844898030545968820545, 7.22972871033631525050809869205, 8.480645719412381995950143571902, 8.792311736872499710709956111953

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