Properties

Label 2-70e2-5.4-c1-0-5
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  + 2.56i·3-s − 3.59·9-s − 1.56·11-s − 5.56i·13-s + 7.16i·17-s + 3.16·19-s + 5.73i·23-s − 1.53i·27-s − 1.96·29-s − 0.969·31-s − 4.03i·33-s + 6.70i·37-s + 14.3·39-s + 8.87·41-s − 4.59i·43-s + ⋯
L(s)  = 1  + 1.48i·3-s − 1.19·9-s − 0.473·11-s − 1.54i·13-s + 1.73i·17-s + 0.726·19-s + 1.19i·23-s − 0.296i·27-s − 0.365·29-s − 0.174·31-s − 0.701i·33-s + 1.10i·37-s + 2.29·39-s + 1.38·41-s − 0.701i·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

\(L(1)\) \(\approx\) \(0.9150596636\)
\(L(\frac12)\) \(\approx\) \(0.9150596636\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.56iT - 3T^{2} \)
11 \( 1 + 1.56T + 11T^{2} \)
13 \( 1 + 5.56iT - 13T^{2} \)
17 \( 1 - 7.16iT - 17T^{2} \)
19 \( 1 - 3.16T + 19T^{2} \)
23 \( 1 - 5.73iT - 23T^{2} \)
29 \( 1 + 1.96T + 29T^{2} \)
31 \( 1 + 0.969T + 31T^{2} \)
37 \( 1 - 6.70iT - 37T^{2} \)
41 \( 1 - 8.87T + 41T^{2} \)
43 \( 1 + 4.59iT - 43T^{2} \)
47 \( 1 - 0.401iT - 47T^{2} \)
53 \( 1 - 9.53iT - 53T^{2} \)
59 \( 1 + 4.56T + 59T^{2} \)
61 \( 1 + 15.3T + 61T^{2} \)
67 \( 1 + 0.862iT - 67T^{2} \)
71 \( 1 + 12.1T + 71T^{2} \)
73 \( 1 + 4iT - 73T^{2} \)
79 \( 1 - 12.8T + 79T^{2} \)
83 \( 1 + 17.1iT - 83T^{2} \)
89 \( 1 + 5.59T + 89T^{2} \)
97 \( 1 - 0.233iT - 97T^{2} \)
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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

−8.879937773987448214540852900665, −7.84788919374237491964397497500, −7.62838882090210976160927425449, −6.14279631841226771664282751762, −5.64841070230459418654915989872, −5.01402846487645541758493749042, −4.17064262775714533885744628111, −3.43880066276524434500235494560, −2.87835861540896966195213799200, −1.41046770498297896636218629033, 0.25108002624822005646855123051, 1.31644423013940276716624265067, 2.28815587614601193031911876857, 2.85406855935742010530346649893, 4.17044657230207210331565710610, 4.97173662176779380863803480056, 5.86330268995660947749282085915, 6.61844428076656750814054566549, 7.17331775187993384222643078682, 7.60719359716237947480977937656

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