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Properties

Label 2-650-25.6-c1-0-12
Degree $2$
Conductor $650$
Sign $0.175 - 0.984i$
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  + (0.309 + 0.951i)2-s + (−0.892 + 0.648i)3-s + (−0.809 + 0.587i)4-s + (−1.63 − 1.52i)5-s + (−0.892 − 0.648i)6-s + 4.94·7-s + (−0.809 − 0.587i)8-s + (−0.550 + 1.69i)9-s + (0.939 − 2.02i)10-s + (−0.745 − 2.29i)11-s + (0.340 − 1.04i)12-s + (0.309 − 0.951i)13-s + (1.52 + 4.70i)14-s + (2.44 + 0.294i)15-s + (0.309 − 0.951i)16-s + (3.90 + 2.83i)17-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (−0.515 + 0.374i)3-s + (−0.404 + 0.293i)4-s + (−0.733 − 0.680i)5-s + (−0.364 − 0.264i)6-s + 1.86·7-s + (−0.286 − 0.207i)8-s + (−0.183 + 0.565i)9-s + (0.297 − 0.641i)10-s + (−0.224 − 0.691i)11-s + (0.0984 − 0.302i)12-s + (0.0857 − 0.263i)13-s + (0.408 + 1.25i)14-s + (0.632 + 0.0759i)15-s + (0.0772 − 0.237i)16-s + (0.947 + 0.688i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.175 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.175 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(650\)    =    \(2 \cdot 5^{2} \cdot 13\)
Sign: $0.175 - 0.984i$
Analytic conductor: \(5.19027\)
Root analytic conductor: \(2.27821\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{650} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 650,\ (\ :1/2),\ 0.175 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08388 + 0.907978i\)
\(L(\frac12)\) \(\approx\) \(1.08388 + 0.907978i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 - 0.951i)T \)
5 \( 1 + (1.63 + 1.52i)T \)
13 \( 1 + (-0.309 + 0.951i)T \)
good3 \( 1 + (0.892 - 0.648i)T + (0.927 - 2.85i)T^{2} \)
7 \( 1 - 4.94T + 7T^{2} \)
11 \( 1 + (0.745 + 2.29i)T + (-8.89 + 6.46i)T^{2} \)
17 \( 1 + (-3.90 - 2.83i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-4.56 - 3.31i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-0.0249 - 0.0767i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (-0.133 + 0.0968i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-4.56 - 3.31i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.24 - 6.90i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.66 - 5.13i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 0.386T + 43T^{2} \)
47 \( 1 + (7.76 - 5.64i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-9.03 + 6.56i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (1.02 - 3.16i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (2.04 + 6.29i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-4.97 - 3.61i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (-4.34 + 3.15i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (1.94 + 5.99i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-8.65 + 6.28i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (4.41 + 3.20i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (1.69 + 5.22i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (6.02 - 4.37i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.89648099609598160863025257506, −9.970214607161200325997661453398, −8.411692625778891281077859368899, −8.199913421482428784264339212482, −7.55280699799264869272678410720, −5.91555291770509784618044056909, −5.11309064018618591695702331261, −4.71491619363023596742524772461, −3.46528479589856441272138558065, −1.29764943033498539059610512879, 0.960861632354637357559499430155, 2.39245494485438277328944800914, 3.72088505849519754537853619918, 4.80490207774270283057660347574, 5.54884806742512074333416664228, 7.00859331776174953787887569502, 7.59593583520846681202390118356, 8.575390618740565091497341445271, 9.697960009973244752385371754614, 10.72211508201745621698772060733

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