Properties

Label 2-1950-13.10-c1-0-20
Degree $2$
Conductor $1950$
Sign $0.376 - 0.926i$
Analytic cond. $15.5708$
Root an. cond. $3.94598$
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  + (−0.866 + 0.5i)2-s + (0.5 + 0.866i)3-s + (0.499 − 0.866i)4-s + (−0.866 − 0.499i)6-s + (3.96 + 2.29i)7-s + 0.999i·8-s + (−0.499 + 0.866i)9-s + (−0.563 + 0.325i)11-s + 0.999·12-s + (1.31 − 3.35i)13-s − 4.58·14-s + (−0.5 − 0.866i)16-s + (1.75 − 3.03i)17-s − 0.999i·18-s + (2.50 + 1.44i)19-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.288 + 0.499i)3-s + (0.249 − 0.433i)4-s + (−0.353 − 0.204i)6-s + (1.49 + 0.865i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s + (−0.169 + 0.0981i)11-s + 0.288·12-s + (0.364 − 0.931i)13-s − 1.22·14-s + (−0.125 − 0.216i)16-s + (0.424 − 0.735i)17-s − 0.235i·18-s + (0.575 + 0.332i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.376 - 0.926i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.376 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $0.376 - 0.926i$
Analytic conductor: \(15.5708\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1950} (751, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ 0.376 - 0.926i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.826318729\)
\(L(\frac12)\) \(\approx\) \(1.826318729\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 \)
13 \( 1 + (-1.31 + 3.35i)T \)
good7 \( 1 + (-3.96 - 2.29i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.563 - 0.325i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.75 + 3.03i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.50 - 1.44i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.42 - 2.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.78 - 8.28i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 5.58iT - 31T^{2} \)
37 \( 1 + (-4.00 + 2.31i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.03 - 1.75i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.85 + 6.68i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 2.08iT - 47T^{2} \)
53 \( 1 + 5.58T + 53T^{2} \)
59 \( 1 + (-6.47 - 3.73i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.31 + 5.74i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.93 - 4.58i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (11.2 + 6.51i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 1.04iT - 73T^{2} \)
79 \( 1 + 3.46T + 79T^{2} \)
83 \( 1 + 10.5iT - 83T^{2} \)
89 \( 1 + (7.10 - 4.09i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-12.5 - 7.21i)T + (48.5 + 84.0i)T^{2} \)
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

−9.070142049805822766402240911063, −8.651418489358986953435256481398, −7.80565802298473159222816001839, −7.41630494744905044338980411232, −5.98198115021039619148849157704, −5.29765759348338490949669427461, −4.76592494280609440889533141062, −3.34996442263727182475483962073, −2.34648961858723105910180877994, −1.16982855535959833501619611293, 0.997749474209282746401490028993, 1.71803743753375664520319158640, 2.86068068724522115340355781195, 4.08378133008971080124391144584, 4.74554851796529701361374120443, 6.05212740347097337543283355149, 6.96256568122999903362164510141, 7.66957124439416963230362955008, 8.273317603005060001641122444834, 8.809430941486423348383739338870

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