↑

Properties

Label 2-950-475.111-c1-0-39
Degree $2$
Conductor $950$
Sign $-0.781 + 0.624i$
Analytic cond. $7.58578$
Root an. cond. $2.75423$
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  + (−0.997 − 0.0697i)2-s + (−0.0816 − 2.33i)3-s + (0.990 + 0.139i)4-s + (1.44 − 1.70i)5-s + (−0.0816 + 2.33i)6-s + (1.27 + 2.21i)7-s + (−0.978 − 0.207i)8-s + (−2.47 + 0.172i)9-s + (−1.56 + 1.59i)10-s + (0.542 + 5.16i)11-s + (0.244 − 2.32i)12-s + (−3.07 − 4.56i)13-s + (−1.11 − 2.29i)14-s + (−4.10 − 3.25i)15-s + (0.961 + 0.275i)16-s + (1.61 − 4.00i)17-s + ⋯
L(s)  = 1  + (−0.705 − 0.0493i)2-s + (−0.0471 − 1.35i)3-s + (0.495 + 0.0695i)4-s + (0.648 − 0.761i)5-s + (−0.0333 + 0.954i)6-s + (0.482 + 0.836i)7-s + (−0.345 − 0.0735i)8-s + (−0.823 + 0.0575i)9-s + (−0.494 + 0.505i)10-s + (0.163 + 1.55i)11-s + (0.0706 − 0.671i)12-s + (−0.853 − 1.26i)13-s + (−0.299 − 0.613i)14-s + (−1.05 − 0.839i)15-s + (0.240 + 0.0689i)16-s + (0.392 − 0.972i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.781 + 0.624i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.781 + 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.781 + 0.624i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.781 + 0.624i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.365601 - 1.04308i\)
\(L(\frac12)\) \(\approx\) \(0.365601 - 1.04308i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.997 + 0.0697i)T \)
5 \( 1 + (-1.44 + 1.70i)T \)
19 \( 1 + (4.30 + 0.654i)T \)
good3 \( 1 + (0.0816 + 2.33i)T + (-2.99 + 0.209i)T^{2} \)
7 \( 1 + (-1.27 - 2.21i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.542 - 5.16i)T + (-10.7 + 2.28i)T^{2} \)
13 \( 1 + (3.07 + 4.56i)T + (-4.86 + 12.0i)T^{2} \)
17 \( 1 + (-1.61 + 4.00i)T + (-12.2 - 11.8i)T^{2} \)
23 \( 1 + (1.09 + 1.06i)T + (0.802 + 22.9i)T^{2} \)
29 \( 1 + (2.20 + 5.45i)T + (-20.8 + 20.1i)T^{2} \)
31 \( 1 + (-5.04 + 5.60i)T + (-3.24 - 30.8i)T^{2} \)
37 \( 1 + (0.980 + 0.712i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (1.10 + 0.316i)T + (34.7 + 21.7i)T^{2} \)
43 \( 1 + (-0.197 + 1.11i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (2.19 + 5.43i)T + (-33.8 + 32.6i)T^{2} \)
53 \( 1 + (7.69 + 1.08i)T + (50.9 + 14.6i)T^{2} \)
59 \( 1 + (0.273 + 1.09i)T + (-52.0 + 27.6i)T^{2} \)
61 \( 1 + (-10.5 - 10.1i)T + (2.12 + 60.9i)T^{2} \)
67 \( 1 + (-11.6 - 7.29i)T + (29.3 + 60.2i)T^{2} \)
71 \( 1 + (6.11 + 3.25i)T + (39.7 + 58.8i)T^{2} \)
73 \( 1 + (-0.723 + 1.07i)T + (-27.3 - 67.6i)T^{2} \)
79 \( 1 + (0.200 + 5.73i)T + (-78.8 + 5.51i)T^{2} \)
83 \( 1 + (-6.87 + 7.63i)T + (-8.67 - 82.5i)T^{2} \)
89 \( 1 + (8.23 - 2.36i)T + (75.4 - 47.1i)T^{2} \)
97 \( 1 + (-15.9 + 9.97i)T + (42.5 - 87.1i)T^{2} \)
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

−9.762236809131979989049691532287, −8.744369397447165670769825640541, −7.966283904212014607600807965182, −7.40304143690412018442965785511, −6.44688326270267029794861258022, −5.54385152251614403908357154674, −4.65314338588653172219538381256, −2.34959613265752219567276607648, −2.09011180479653844553955224020, −0.64458551766055438804864854733, 1.65363287118991811414339874357, 3.16877933618777099410572560463, 4.02526656374315987226327099383, 5.10547773502170579225594607195, 6.20468547623585681559698014258, 6.91091300920144348306535295786, 8.090462830783119985109023354805, 8.894652877887430744858367048198, 9.706919899840849973713947135773, 10.29450982853789977234824985873

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