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Properties

Label 2-1300-65.49-c1-0-15
Degree $2$
Conductor $1300$
Sign $0.516 + 0.856i$
Analytic cond. $10.3805$
Root an. cond. $3.22188$
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  + (2.44 − 1.41i)3-s + (−1.04 + 1.81i)7-s + (2.49 − 4.32i)9-s + (1.5 − 0.866i)11-s + (−0.331 − 3.59i)13-s + (3.14 + 1.81i)17-s + (−0.926 − 0.534i)19-s + 5.92i·21-s + (6.77 − 3.90i)23-s − 5.62i·27-s + (−0.263 − 0.456i)29-s − 5.84i·31-s + (2.44 − 4.24i)33-s + (−4.87 − 8.44i)37-s + (−5.88 − 8.32i)39-s + ⋯
L(s)  = 1  + (1.41 − 0.816i)3-s + (−0.395 + 0.685i)7-s + (0.831 − 1.44i)9-s + (0.452 − 0.261i)11-s + (−0.0918 − 0.995i)13-s + (0.762 + 0.439i)17-s + (−0.212 − 0.122i)19-s + 1.29i·21-s + (1.41 − 0.815i)23-s − 1.08i·27-s + (−0.0489 − 0.0847i)29-s − 1.04i·31-s + (0.426 − 0.738i)33-s + (−0.801 − 1.38i)37-s + (−0.942 − 1.33i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.516 + 0.856i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.516 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1300\)    =    \(2^{2} \cdot 5^{2} \cdot 13\)
Sign: $0.516 + 0.856i$
Analytic conductor: \(10.3805\)
Root analytic conductor: \(3.22188\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1300} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1300,\ (\ :1/2),\ 0.516 + 0.856i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.714834393\)
\(L(\frac12)\) \(\approx\) \(2.714834393\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13 \( 1 + (0.331 + 3.59i)T \)
good3 \( 1 + (-2.44 + 1.41i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (1.04 - 1.81i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.5 + 0.866i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-3.14 - 1.81i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.926 + 0.534i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-6.77 + 3.90i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.263 + 0.456i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 5.84iT - 31T^{2} \)
37 \( 1 + (4.87 + 8.44i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.69 - 2.13i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-8.09 - 4.67i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 3.46T + 47T^{2} \)
53 \( 1 - 12.5iT - 53T^{2} \)
59 \( 1 + (-1.21 - 0.701i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.55 + 9.62i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.41 - 9.38i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (12.2 + 7.08i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 2.64T + 73T^{2} \)
79 \( 1 + 13.5T + 79T^{2} \)
83 \( 1 + 15.7T + 83T^{2} \)
89 \( 1 + (-4.78 + 2.76i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.59 - 13.1i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.102261772454638582154733805327, −8.825455208045696854835000111897, −7.898108612358166600184350254309, −7.31848383078671951689203882503, −6.32295537995586847624337166770, −5.50128502963592439403919233264, −4.02467202922743716353263963720, −3.02940478931307364545516686689, −2.45759816110481742639502518202, −1.07826631802739986086674500724, 1.55313872269352465061618163427, 2.90605863736703013975558169121, 3.62929623649893108948235571052, 4.37535909184439234691469129446, 5.33566533600925463933327075948, 6.93945162705227488756018467263, 7.22692997394872676028634125286, 8.501890461454698207538664042393, 8.927964087177345271943800579117, 9.824303568696996131457909806228

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