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Properties

Label 2-650-65.29-c1-0-0
Degree $2$
Conductor $650$
Sign $-0.658 + 0.752i$
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 + (−1.99 + 1.15i)3-s + (0.499 − 0.866i)4-s + (1.15 − 1.99i)6-s + (0.603 + 0.348i)7-s + 0.999i·8-s + (1.15 − 1.99i)9-s + (0.348 + 0.603i)11-s + 2.30i·12-s + (−3.12 + 1.80i)13-s − 0.697·14-s + (−0.5 − 0.866i)16-s + (2.51 + 1.45i)17-s + 2.30i·18-s + (−0.197 + 0.341i)19-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−1.15 + 0.664i)3-s + (0.249 − 0.433i)4-s + (0.470 − 0.814i)6-s + (0.228 + 0.131i)7-s + 0.353i·8-s + (0.383 − 0.664i)9-s + (0.105 + 0.182i)11-s + 0.664i·12-s + (−0.866 + 0.499i)13-s − 0.186·14-s + (−0.125 − 0.216i)16-s + (0.610 + 0.352i)17-s + 0.542i·18-s + (−0.0452 + 0.0783i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.658 + 0.752i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.658 + 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(650\)    =    \(2 \cdot 5^{2} \cdot 13\)
Sign: $-0.658 + 0.752i$
Analytic conductor: \(5.19027\)
Root analytic conductor: \(2.27821\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{650} (549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 650,\ (\ :1/2),\ -0.658 + 0.752i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0465425 - 0.102596i\)
\(L(\frac12)\) \(\approx\) \(0.0465425 - 0.102596i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.12 - 1.80i)T \)
good3 \( 1 + (1.99 - 1.15i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (-0.603 - 0.348i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.348 - 0.603i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.51 - 1.45i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.197 - 0.341i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.85 - 2.80i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.65 - 2.86i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 8.60T + 31T^{2} \)
37 \( 1 + (-3.20 + 1.84i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.5 + 0.866i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (7.37 + 4.25i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 10.3iT - 47T^{2} \)
53 \( 1 + 12.2iT - 53T^{2} \)
59 \( 1 + (5.10 - 8.84i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.10 - 3.64i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.85 - 2.80i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.30 + 9.18i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 8iT - 73T^{2} \)
79 \( 1 + 9.30T + 79T^{2} \)
83 \( 1 - 16.8iT - 83T^{2} \)
89 \( 1 + (-0.0458 - 0.0793i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (10.1 + 5.84i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.94846529698751260310507926371, −10.15395366253292508290144938254, −9.628069624788541997730972762162, −8.559032210168798517841525244070, −7.53296396892150474325904328277, −6.63571245788178473617157325143, −5.60124771807485113457977400203, −5.03394124440056408489309937750, −3.82551392903124168356635352041, −1.89695133153836001977767962087, 0.087509765701600440654719947573, 1.45082208913571055841775943188, 2.93159717839683273713726441280, 4.51031613078481387862183314941, 5.62945784372776124684314319931, 6.44676966278357932119792870065, 7.45431086020353706924004101354, 8.037915678333098284255392065102, 9.323864449564156019801611873778, 10.12116357395872535396397676885

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