↑

Properties

Label 2-1300-65.29-c1-0-20
Degree $2$
Conductor $1300$
Sign $-0.458 + 0.888i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (2.59 − 1.5i)3-s + (−2.59 − 1.5i)7-s + (3 − 5.19i)9-s + (−1.5 − 2.59i)11-s + (−3.46 − i)13-s + (6.06 + 3.5i)17-s + (0.5 − 0.866i)19-s − 9·21-s + (6.06 − 3.5i)23-s − 9i·27-s + (−2.5 − 4.33i)29-s − 4·31-s + (−7.79 − 4.5i)33-s + (−2.59 + 1.5i)37-s + (−10.5 + 2.59i)39-s + ⋯
L(s)  = 1  + (1.49 − 0.866i)3-s + (−0.981 − 0.566i)7-s + (1 − 1.73i)9-s + (−0.452 − 0.783i)11-s + (−0.960 − 0.277i)13-s + (1.47 + 0.848i)17-s + (0.114 − 0.198i)19-s − 1.96·21-s + (1.26 − 0.729i)23-s − 1.73i·27-s + (−0.464 − 0.804i)29-s − 0.718·31-s + (−1.35 − 0.783i)33-s + (−0.427 + 0.246i)37-s + (−1.68 + 0.416i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.458 + 0.888i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.458 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1300\)    =    \(2^{2} \cdot 5^{2} \cdot 13\)
Sign: $-0.458 + 0.888i$
Analytic conductor: \(10.3805\)
Root analytic conductor: \(3.22188\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1300} (549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1300,\ (\ :1/2),\ -0.458 + 0.888i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.217848770\)
\(L(\frac12)\) \(\approx\) \(2.217848770\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.46 + i)T \)
good3 \( 1 + (-2.59 + 1.5i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (2.59 + 1.5i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-6.06 - 3.5i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-6.06 + 3.5i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.5 + 4.33i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (2.59 - 1.5i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.5 + 6.06i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (7.79 + 4.5i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 8iT - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 + (-2.5 + 4.33i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.5 + 4.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-11.2 + 6.5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.5 + 2.59i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 14iT - 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + 12iT - 83T^{2} \)
89 \( 1 + (-3.5 - 6.06i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-9.52 - 5.5i)T + (48.5 + 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.302767612793782828129742105811, −8.449886275596824471331505137575, −7.77974349741270454721636439108, −7.15039373005845966717277190854, −6.37098438775061410126775251949, −5.20768396828488599641568330479, −3.60297079475479156408754319653, −3.23359060522149954992502521669, −2.19566575403911816895561774177, −0.74415245683020832165539587452, 2.01070486842871243403993685576, 3.07715631262917029914693650383, 3.43849065094132470166468523251, 4.82997563268097608512859044859, 5.38437232919076095023508226281, 7.01543148840224216825071019070, 7.49813233681921234583556075093, 8.499147717110062427808695341904, 9.280805446374085115734139952406, 9.839959566340699410845550903493

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