Properties

Label 2-70e2-5.4-c1-0-40
Degree $2$
Conductor $4900$
Sign $0.894 - 0.447i$
Analytic cond. $39.1266$
Root an. cond. $6.25513$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.20i·3-s − 7.26·9-s + 4.20·11-s − 0.204i·13-s − 5.06i·17-s − 1.06·19-s − 2.14i·23-s − 13.6i·27-s + 7.47·29-s − 8.47·31-s + 13.4i·33-s − 10.6i·37-s + 0.654·39-s + 10.5·41-s − 8.26i·43-s + ⋯
L(s)  = 1  + 1.85i·3-s − 2.42·9-s + 1.26·11-s − 0.0566i·13-s − 1.22i·17-s − 0.244·19-s − 0.446i·23-s − 2.63i·27-s + 1.38·29-s − 1.52·31-s + 2.34i·33-s − 1.74i·37-s + 0.104·39-s + 1.64·41-s − 1.26i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4900\)    =    \(2^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(39.1266\)
Root analytic conductor: \(6.25513\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4900} (2549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4900,\ (\ :1/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.680511690\)
\(L(\frac12)\) \(\approx\) \(1.680511690\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 - 3.20iT - 3T^{2} \)
11 \( 1 - 4.20T + 11T^{2} \)
13 \( 1 + 0.204iT - 13T^{2} \)
17 \( 1 + 5.06iT - 17T^{2} \)
19 \( 1 + 1.06T + 19T^{2} \)
23 \( 1 + 2.14iT - 23T^{2} \)
29 \( 1 - 7.47T + 29T^{2} \)
31 \( 1 + 8.47T + 31T^{2} \)
37 \( 1 + 10.6iT - 37T^{2} \)
41 \( 1 - 10.5T + 41T^{2} \)
43 \( 1 + 8.26iT - 43T^{2} \)
47 \( 1 - 3.26iT - 47T^{2} \)
53 \( 1 + 5.67iT - 53T^{2} \)
59 \( 1 + 1.20T + 59T^{2} \)
61 \( 1 - 1.65T + 61T^{2} \)
67 \( 1 + 12.4iT - 67T^{2} \)
71 \( 1 + 0.591T + 71T^{2} \)
73 \( 1 - 4iT - 73T^{2} \)
79 \( 1 + 6.54T + 79T^{2} \)
83 \( 1 + 3.88iT - 83T^{2} \)
89 \( 1 - 9.26T + 89T^{2} \)
97 \( 1 - 1.33iT - 97T^{2} \)
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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

−8.746900093912928959161368731362, −7.68146887778372993318185537527, −6.77750225077649449919210101107, −5.92182810563705022663516987894, −5.24050142291693700563596345649, −4.48094701931861056405585724120, −3.93879652215002282615838481353, −3.22199728399969216654504928126, −2.26269684603696828999613395282, −0.51764563834204690497393570321, 1.06546267104759888255676166185, 1.58643510802582309113044729495, 2.56234428926726171154820113367, 3.50248787243601049869868613782, 4.47814563463511691880672931704, 5.69642559945680898051462557595, 6.25293967376306435430030522584, 6.71833750458807074697128821451, 7.43964325413245013672623386005, 8.111939710080627193114012497011

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