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Properties

Label 2-650-13.10-c1-0-19
Degree $2$
Conductor $650$
Sign $-0.875 + 0.484i$
Analytic cond. $5.19027$
Root an. cond. $2.27821$
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  + (0.866 − 0.5i)2-s + (−0.609 − 1.05i)3-s + (0.499 − 0.866i)4-s + (−1.05 − 0.609i)6-s + (−3.07 − 1.77i)7-s − 0.999i·8-s + (0.756 − 1.30i)9-s + (4.22 − 2.44i)11-s − 1.21·12-s + (−0.0197 + 3.60i)13-s − 3.55·14-s + (−0.5 − 0.866i)16-s + (−1.57 + 2.72i)17-s − 1.51i·18-s + (−6.90 − 3.98i)19-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.352 − 0.609i)3-s + (0.249 − 0.433i)4-s + (−0.431 − 0.248i)6-s + (−1.16 − 0.671i)7-s − 0.353i·8-s + (0.252 − 0.436i)9-s + (1.27 − 0.736i)11-s − 0.352·12-s + (−0.00548 + 0.999i)13-s − 0.949·14-s + (−0.125 − 0.216i)16-s + (−0.382 + 0.662i)17-s − 0.356i·18-s + (−1.58 − 0.914i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.875 + 0.484i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.875 + 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(650\)    =    \(2 \cdot 5^{2} \cdot 13\)
Sign: $-0.875 + 0.484i$
Analytic conductor: \(5.19027\)
Root analytic conductor: \(2.27821\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{650} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 650,\ (\ :1/2),\ -0.875 + 0.484i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.357615 - 1.38524i\)
\(L(\frac12)\) \(\approx\) \(0.357615 - 1.38524i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 \)
13 \( 1 + (0.0197 - 3.60i)T \)
good3 \( 1 + (0.609 + 1.05i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (3.07 + 1.77i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-4.22 + 2.44i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.57 - 2.72i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (6.90 + 3.98i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.45 + 4.24i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.02 + 3.51i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 5.02iT - 31T^{2} \)
37 \( 1 + (-1.82 + 1.05i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.49 + 3.17i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.922 + 1.59i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 6.08iT - 47T^{2} \)
53 \( 1 - 8.08T + 53T^{2} \)
59 \( 1 + (-10.5 - 6.11i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.09 - 3.62i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-13.5 + 7.83i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-6.33 - 3.65i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 5.54iT - 73T^{2} \)
79 \( 1 + 2.01T + 79T^{2} \)
83 \( 1 - 6.23iT - 83T^{2} \)
89 \( 1 + (5.30 - 3.06i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.97 - 1.13i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.37907348356608486598633188760, −9.377050673931338246758632957103, −8.638555653040214798845075865758, −6.85076157901465610487621363408, −6.69183965188543146240946520535, −5.96877241121600282494624155160, −4.15614185185765703040511591601, −3.82614124293677465906625981845, −2.16882771281037183022752813944, −0.64708508511144683777209414801, 2.24585247517487369790356068192, 3.64053781943146960982825143477, 4.40040880450975650435928456098, 5.56994383807101698520718351559, 6.23094579021835089774937250646, 7.16230827875569036266735173775, 8.251616865367108329657020936044, 9.438942617314975393582415172871, 9.898157253518658238217275080221, 10.97768631737685941722072126061

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