↑

Properties

Label 2-650-13.9-c1-0-15
Degree $2$
Conductor $650$
Sign $-0.522 + 0.852i$
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 + (−0.499 − 0.866i)4-s + (−1.5 − 2.59i)7-s − 0.999·8-s + (1.5 + 2.59i)9-s + (1.5 − 2.59i)11-s + (3.5 − 0.866i)13-s − 3·14-s + (−0.5 + 0.866i)16-s + (−2 − 3.46i)17-s + 3·18-s + (−3.5 − 6.06i)19-s + (−1.5 − 2.59i)22-s + (−2 + 3.46i)23-s + (1 − 3.46i)26-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.566 − 0.981i)7-s − 0.353·8-s + (0.5 + 0.866i)9-s + (0.452 − 0.783i)11-s + (0.970 − 0.240i)13-s − 0.801·14-s + (−0.125 + 0.216i)16-s + (−0.485 − 0.840i)17-s + 0.707·18-s + (−0.802 − 1.39i)19-s + (−0.319 − 0.553i)22-s + (−0.417 + 0.722i)23-s + (0.196 − 0.679i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.522 + 0.852i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.522 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(650\)    =    \(2 \cdot 5^{2} \cdot 13\)
Sign: $-0.522 + 0.852i$
Analytic conductor: \(5.19027\)
Root analytic conductor: \(2.27821\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{650} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 650,\ (\ :1/2),\ -0.522 + 0.852i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.755677 - 1.34851i\)
\(L(\frac12)\) \(\approx\) \(0.755677 - 1.34851i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + (1.5 + 2.59i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (2 + 3.46i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.5 + 6.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4 + 6.92i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 10T + 31T^{2} \)
37 \( 1 + (-1.5 + 2.59i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1 + 1.73i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3 - 5.19i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - T + 47T^{2} \)
53 \( 1 - 9T + 53T^{2} \)
59 \( 1 + (-2 - 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7 - 12.1i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3 + 5.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 4T + 73T^{2} \)
79 \( 1 - 10T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (0.5 - 0.866i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-8 - 13.8i)T + (-48.5 + 84.0i)T^{2} \)
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

−10.56137925077447357653295631519, −9.518812864310649863309719700539, −8.721345803051156553388781856967, −7.52647273559935246481970762978, −6.65890804551353969216972077966, −5.62798750014615608750309659937, −4.39678740192933966420872304385, −3.69638476857268843424200575725, −2.39196469143154050840628260786, −0.77054046507760470462636308043, 1.88264824026137694226345291465, 3.55952613992839848789754558056, 4.23378446391115973481511762787, 5.68084442415512962177400840998, 6.36433241489654664088541713359, 6.99717128473903798836486865976, 8.401474518658479747614416471365, 8.913744528232178928071515871359, 9.812177129001117452654268639964, 10.79017168823907123621988358100

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