↑

Properties

Label 2-650-65.2-c1-0-7
Degree $2$
Conductor $650$
Sign $0.955 + 0.293i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)2-s + (−2.36 − 0.633i)3-s + (−0.499 + 0.866i)4-s + (−0.633 − 2.36i)6-s + (0.232 + 0.133i)7-s − 0.999·8-s + (2.59 + 1.50i)9-s + (0.133 − 0.5i)11-s + (1.73 − 1.73i)12-s + (−3.5 − 0.866i)13-s + 0.267i·14-s + (−0.5 − 0.866i)16-s + (0.0980 + 0.366i)17-s + 3.00i·18-s + (7.33 − 1.96i)19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−1.36 − 0.366i)3-s + (−0.249 + 0.433i)4-s + (−0.258 − 0.965i)6-s + (0.0877 + 0.0506i)7-s − 0.353·8-s + (0.866 + 0.500i)9-s + (0.0403 − 0.150i)11-s + (0.499 − 0.499i)12-s + (−0.970 − 0.240i)13-s + 0.0716i·14-s + (−0.125 − 0.216i)16-s + (0.0237 + 0.0887i)17-s + 0.707i·18-s + (1.68 − 0.450i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 + 0.293i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.955 + 0.293i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(650\)    =    \(2 \cdot 5^{2} \cdot 13\)
Sign: $0.955 + 0.293i$
Analytic conductor: \(5.19027\)
Root analytic conductor: \(2.27821\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{650} (457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 650,\ (\ :1/2),\ 0.955 + 0.293i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.945636 - 0.142111i\)
\(L(\frac12)\) \(\approx\) \(0.945636 - 0.142111i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 \)
13 \( 1 + (3.5 + 0.866i)T \)
good3 \( 1 + (2.36 + 0.633i)T + (2.59 + 1.5i)T^{2} \)
7 \( 1 + (-0.232 - 0.133i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.133 + 0.5i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-0.0980 - 0.366i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-7.33 + 1.96i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-1.63 + 6.09i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-2.19 + 1.26i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3 + 3i)T - 31iT^{2} \)
37 \( 1 + (0.232 - 0.133i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.09 - 1.36i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-8.19 + 2.19i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + 7iT - 47T^{2} \)
53 \( 1 + (-4.63 + 4.63i)T - 53iT^{2} \)
59 \( 1 + (-2.36 - 8.83i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (4.63 - 8.02i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.46 - 11.1i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.53 + 13.1i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + 1.26T + 73T^{2} \)
79 \( 1 + 13.1iT - 79T^{2} \)
83 \( 1 - 0.732iT - 83T^{2} \)
89 \( 1 + (16.8 + 4.52i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (2.90 - 5.02i)T + (-48.5 - 84.0i)T^{2} \)
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

−10.63495902214581799420779662757, −9.752499934502179238799145345645, −8.619225790006694402680770105892, −7.46913155425405006885976855429, −6.89938912804763602235387200778, −5.90286706201570622949760824796, −5.25236527439691282315599646261, −4.40672536573058219928309505641, −2.77327367001910599359441083788, −0.66748993917063844112725725487, 1.16120969516472302350362493471, 2.92033387793325913667571984533, 4.25535177197620980592831021498, 5.13706276972348435608376928992, 5.70190339991572958621522657867, 6.85038772076547522645035960388, 7.81799823746754683764975526430, 9.461778256551768247536488553696, 9.791790466098884307592549934929, 10.88575936987294511427136727062

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