↑

Properties

Label 2-950-19.7-c1-0-23
Degree $2$
Conductor $950$
Sign $0.431 + 0.901i$
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  + (−0.5 − 0.866i)2-s + (0.741 + 1.28i)3-s + (−0.499 + 0.866i)4-s + (0.741 − 1.28i)6-s + 0.482·7-s + 0.999·8-s + (0.400 − 0.693i)9-s − 4.43·11-s − 1.48·12-s + (2.07 − 3.58i)13-s + (−0.241 − 0.418i)14-s + (−0.5 − 0.866i)16-s + (−3.94 − 6.84i)17-s − 0.801·18-s + (4.31 − 0.590i)19-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.428 + 0.741i)3-s + (−0.249 + 0.433i)4-s + (0.302 − 0.524i)6-s + 0.182·7-s + 0.353·8-s + (0.133 − 0.231i)9-s − 1.33·11-s − 0.428·12-s + (0.574 − 0.994i)13-s + (−0.0645 − 0.111i)14-s + (−0.125 − 0.216i)16-s + (−0.957 − 1.65i)17-s − 0.188·18-s + (0.990 − 0.135i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.431 + 0.901i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.431 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.431 + 0.901i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (501, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.431 + 0.901i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16233 - 0.732144i\)
\(L(\frac12)\) \(\approx\) \(1.16233 - 0.732144i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19 \( 1 + (-4.31 + 0.590i)T \)
good3 \( 1 + (-0.741 - 1.28i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 - 0.482T + 7T^{2} \)
11 \( 1 + 4.43T + 11T^{2} \)
13 \( 1 + (-2.07 + 3.58i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.94 + 6.84i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-2.82 + 4.90i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.91 - 3.31i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 7.58T + 31T^{2} \)
37 \( 1 - 3.50T + 37T^{2} \)
41 \( 1 + (-4.44 - 7.70i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.318 + 0.551i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.13 - 3.69i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.59 + 2.76i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.812 + 1.40i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.735 - 1.27i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.59 + 11.4i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.41 + 2.44i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (3.43 + 5.95i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.576 - 0.997i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 12.4T + 83T^{2} \)
89 \( 1 + (4.37 - 7.58i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.476 + 0.825i)T + (-48.5 + 84.0i)T^{2} \)
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

−9.806333494128020400173536541732, −9.302846953267250159650088315700, −8.361980301158609172963175233556, −7.69669008371599175685334717375, −6.57649345071213982663330957464, −5.11358553358361030698461036763, −4.57668300721514446930735031087, −3.14335981894938167360761445461, −2.71873771090678611083702434964, −0.74665402422251132205667478924, 1.41922755625064251589519967907, 2.46196984709624954145449595615, 4.01438021442168995116662723966, 5.11062816817042781290695704041, 6.09478479609371042436011636936, 6.96436591057964004487705529773, 7.78123221747833576407615706770, 8.270557306341265599896474512643, 9.101485323944194487533197945108, 10.12158419986959192931696458462

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