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Properties

Label 2-950-25.6-c1-0-10
Degree $2$
Conductor $950$
Sign $0.250 - 0.968i$
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  + (−0.309 − 0.951i)2-s + (−0.444 + 0.322i)3-s + (−0.809 + 0.587i)4-s + (−0.351 + 2.20i)5-s + (0.444 + 0.322i)6-s + 4.87·7-s + (0.809 + 0.587i)8-s + (−0.833 + 2.56i)9-s + (2.20 − 0.347i)10-s + (0.198 + 0.611i)11-s + (0.169 − 0.522i)12-s + (−0.844 + 2.59i)13-s + (−1.50 − 4.63i)14-s + (−0.556 − 1.09i)15-s + (0.309 − 0.951i)16-s + (−2.11 − 1.53i)17-s + ⋯
L(s)  = 1  + (−0.218 − 0.672i)2-s + (−0.256 + 0.186i)3-s + (−0.404 + 0.293i)4-s + (−0.157 + 0.987i)5-s + (0.181 + 0.131i)6-s + 1.84·7-s + (0.286 + 0.207i)8-s + (−0.277 + 0.855i)9-s + (0.698 − 0.109i)10-s + (0.0599 + 0.184i)11-s + (0.0489 − 0.150i)12-s + (−0.234 + 0.720i)13-s + (−0.402 − 1.23i)14-s + (−0.143 − 0.282i)15-s + (0.0772 − 0.237i)16-s + (−0.511 − 0.371i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.250 - 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.250 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.250 - 0.968i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (381, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.250 - 0.968i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.934222 + 0.723206i\)
\(L(\frac12)\) \(\approx\) \(0.934222 + 0.723206i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.309 + 0.951i)T \)
5 \( 1 + (0.351 - 2.20i)T \)
19 \( 1 + (0.809 + 0.587i)T \)
good3 \( 1 + (0.444 - 0.322i)T + (0.927 - 2.85i)T^{2} \)
7 \( 1 - 4.87T + 7T^{2} \)
11 \( 1 + (-0.198 - 0.611i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (0.844 - 2.59i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (2.11 + 1.53i)T + (5.25 + 16.1i)T^{2} \)
23 \( 1 + (-1.00 - 3.10i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (4.84 - 3.52i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (2.95 + 2.14i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.434 - 1.33i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.454 + 1.40i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 4.74T + 43T^{2} \)
47 \( 1 + (-4.24 + 3.08i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (11.7 - 8.50i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (1.15 - 3.55i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-2.54 - 7.82i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-8.60 - 6.25i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (-0.817 + 0.593i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-1.90 - 5.85i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-8.43 + 6.12i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (2.24 + 1.63i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (-0.558 - 1.71i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (0.463 - 0.336i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.58899300527639672813154285624, −9.482656257555557056398362001633, −8.592125562044923764247907214071, −7.67081978599337667848889217038, −7.17368433723430570569560643324, −5.66184578573811270718593348553, −4.78840234952730696021100389113, −4.01052312428802536514428160187, −2.51406135599459899936001923434, −1.74701311501563607784721420069, 0.63028735750507332786647109571, 1.84952455820277918107581216439, 3.87161479705189959070608308233, 4.83608534865941546544012040921, 5.45120978433289403486496260874, 6.36947056099309866466434079853, 7.60164847035194393670404865658, 8.152258836683748392136473143674, 8.799523449018110060501768531662, 9.562198542840580364293068260901

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