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Properties

Label 2-950-95.12-c1-0-9
Degree $2$
Conductor $950$
Sign $-0.0189 - 0.999i$
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  + (0.965 + 0.258i)2-s + (−0.258 + 0.965i)3-s + (0.866 + 0.499i)4-s + (−0.499 + 0.866i)6-s + (0.707 + 0.707i)8-s + (1.73 + i)9-s − 3·11-s + (−0.707 + 0.707i)12-s + (1.93 − 0.517i)13-s + (0.500 + 0.866i)16-s + (−1.79 + 6.69i)17-s + (1.41 + 1.41i)18-s + (4.33 + 0.5i)19-s + (−2.89 − 0.776i)22-s + (0.896 + 3.34i)23-s + (−0.866 + 0.500i)24-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (−0.149 + 0.557i)3-s + (0.433 + 0.249i)4-s + (−0.204 + 0.353i)6-s + (0.249 + 0.249i)8-s + (0.577 + 0.333i)9-s − 0.904·11-s + (−0.204 + 0.204i)12-s + (0.535 − 0.143i)13-s + (0.125 + 0.216i)16-s + (−0.434 + 1.62i)17-s + (0.333 + 0.333i)18-s + (0.993 + 0.114i)19-s + (−0.617 − 0.165i)22-s + (0.186 + 0.697i)23-s + (−0.176 + 0.102i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0189 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0189 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.0189 - 0.999i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.0189 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.58079 + 1.61106i\)
\(L(\frac12)\) \(\approx\) \(1.58079 + 1.61106i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 - 0.258i)T \)
5 \( 1 \)
19 \( 1 + (-4.33 - 0.5i)T \)
good3 \( 1 + (0.258 - 0.965i)T + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + 7iT^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 + (-1.93 + 0.517i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (1.79 - 6.69i)T + (-14.7 - 8.5i)T^{2} \)
23 \( 1 + (-0.896 - 3.34i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-3.46 + 6i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 3.46iT - 31T^{2} \)
37 \( 1 + (1.41 - 1.41i)T - 37iT^{2} \)
41 \( 1 + (1.5 - 0.866i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.34 - 0.896i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (6.69 - 1.79i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (5.79 - 1.55i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-2.59 - 4.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4 + 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.258 + 0.965i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (3 - 1.73i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-15.0 - 4.03i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (3.46 + 6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.22 + 1.22i)T - 83iT^{2} \)
89 \( 1 + (-3.46 + 6i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.965 - 0.258i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.37262105126046852249812838862, −9.640721534878595459271586752105, −8.367053774969078274616492361744, −7.75894827774093763759394630901, −6.70283989701015241879054895616, −5.74740809306639643708766654918, −4.99070379375697101868200501350, −4.08681416099271967034881855626, −3.18733164728691932726451071036, −1.73903584322564646044460279254, 0.908432586301524351069000868815, 2.35904662918243664188462871579, 3.37147204658133998665261905142, 4.63059588494233870514504629684, 5.34582729375733012930045409672, 6.47777291790842624873453384916, 7.10897355324422826647527665553, 7.896451571325744393889393161948, 9.107378344196652695209215401686, 9.911491333703551601081414846892

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