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Properties

Label 2-950-19.5-c1-0-24
Degree $2$
Conductor $950$
Sign $-0.724 + 0.689i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.939 + 0.342i)2-s + (0.260 − 1.47i)3-s + (0.766 − 0.642i)4-s + (0.260 + 1.47i)6-s + (−1.51 − 2.61i)7-s + (−0.500 + 0.866i)8-s + (0.699 + 0.254i)9-s + (2.53 − 4.39i)11-s + (−0.750 − 1.30i)12-s + (−0.272 − 1.54i)13-s + (2.31 + 1.94i)14-s + (0.173 − 0.984i)16-s + (−1.18 + 0.431i)17-s − 0.744·18-s + (−3.72 + 2.26i)19-s + ⋯
L(s)  = 1  + (−0.664 + 0.241i)2-s + (0.150 − 0.853i)3-s + (0.383 − 0.321i)4-s + (0.106 + 0.603i)6-s + (−0.571 − 0.990i)7-s + (−0.176 + 0.306i)8-s + (0.233 + 0.0848i)9-s + (0.764 − 1.32i)11-s + (−0.216 − 0.375i)12-s + (−0.0756 − 0.428i)13-s + (0.619 + 0.519i)14-s + (0.0434 − 0.246i)16-s + (−0.287 + 0.104i)17-s − 0.175·18-s + (−0.854 + 0.519i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.724 + 0.689i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.724 + 0.689i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.724 + 0.689i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (651, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.724 + 0.689i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.351022 - 0.878472i\)
\(L(\frac12)\) \(\approx\) \(0.351022 - 0.878472i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.939 - 0.342i)T \)
5 \( 1 \)
19 \( 1 + (3.72 - 2.26i)T \)
good3 \( 1 + (-0.260 + 1.47i)T + (-2.81 - 1.02i)T^{2} \)
7 \( 1 + (1.51 + 2.61i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.53 + 4.39i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.272 + 1.54i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (1.18 - 0.431i)T + (13.0 - 10.9i)T^{2} \)
23 \( 1 + (0.795 - 0.667i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (-1.90 - 0.694i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (-0.312 - 0.541i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 5.24T + 37T^{2} \)
41 \( 1 + (0.189 - 1.07i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (8.82 + 7.40i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (9.85 + 3.58i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (-0.484 + 0.406i)T + (9.20 - 52.1i)T^{2} \)
59 \( 1 + (-5.25 + 1.91i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (8.87 - 7.44i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-11.5 - 4.20i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (10.5 + 8.84i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (2.46 - 13.9i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (-0.424 + 2.41i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (1.10 + 1.92i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.09 + 11.9i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (11.4 - 4.17i)T + (74.3 - 62.3i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.807908800163305045991534142201, −8.577818823495638091653090674293, −8.162170392817938541325152945680, −7.09132893322110108153978912056, −6.62147012494279217510469170466, −5.79754031973591300092919313579, −4.23372097308277228055403482708, −3.16808151339233488383145690669, −1.67202009178266081505849481930, −0.54427386359147918379593096196, 1.80132878528720900786721769218, 2.90282921457049724013951980274, 4.13202807857415797791514690556, 4.83965801124919139921764327728, 6.36409733489934647894730443454, 6.86153143863174136609577722519, 8.115613074380143793655452133863, 9.086715422262549868903972651016, 9.503821091580352874340988464027, 10.01667559382645221725071394509

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