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Properties

Label 2-950-95.8-c1-0-27
Degree $2$
Conductor $950$
Sign $-0.911 + 0.411i$
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 + (−0.672 − 2.51i)3-s + (0.866 − 0.499i)4-s + (−1.29 − 2.25i)6-s + (0.692 − 0.692i)7-s + (0.707 − 0.707i)8-s + (−3.25 + 1.87i)9-s + 4.06·11-s + (−1.83 − 1.83i)12-s + (−6.19 − 1.66i)13-s + (0.489 − 0.847i)14-s + (0.500 − 0.866i)16-s + (−0.670 − 2.50i)17-s + (−2.65 + 2.65i)18-s + (−0.843 − 4.27i)19-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (−0.388 − 1.44i)3-s + (0.433 − 0.249i)4-s + (−0.530 − 0.918i)6-s + (0.261 − 0.261i)7-s + (0.249 − 0.249i)8-s + (−1.08 + 0.625i)9-s + 1.22·11-s + (−0.530 − 0.530i)12-s + (−1.71 − 0.460i)13-s + (0.130 − 0.226i)14-s + (0.125 − 0.216i)16-s + (−0.162 − 0.606i)17-s + (−0.625 + 0.625i)18-s + (−0.193 − 0.981i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.911 + 0.411i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.911 + 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.911 + 0.411i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.911 + 0.411i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.398403 - 1.85217i\)
\(L(\frac12)\) \(\approx\) \(0.398403 - 1.85217i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.843 + 4.27i)T \)
good3 \( 1 + (0.672 + 2.51i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (-0.692 + 0.692i)T - 7iT^{2} \)
11 \( 1 - 4.06T + 11T^{2} \)
13 \( 1 + (6.19 + 1.66i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (0.670 + 2.50i)T + (-14.7 + 8.5i)T^{2} \)
23 \( 1 + (-0.943 + 3.52i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-1.89 - 3.28i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 5.30iT - 31T^{2} \)
37 \( 1 + (-2.58 - 2.58i)T + 37iT^{2} \)
41 \( 1 + (7.09 + 4.09i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.61 - 1.23i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (-6.29 - 1.68i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-2.10 - 0.565i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-2.99 + 5.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.43 + 7.68i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.47 - 9.25i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-6.72 - 3.88i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-15.0 + 4.02i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-3.49 + 6.05i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.44 + 4.44i)T + 83iT^{2} \)
89 \( 1 + (1.61 + 2.80i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-9.80 + 2.62i)T + (84.0 - 48.5i)T^{2} \)
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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

−9.776538016650235644811894276698, −8.726740919398340590753584542057, −7.60553955345638069046009283441, −6.90451097852035766658620100282, −6.53415319243074325784887326135, −5.24910554729063751266996095247, −4.54981107563826516395433067724, −3.00273016623014618321349551564, −1.98903742056203245438706820229, −0.73422037487866606701616478460, 2.10127601947059978195804069023, 3.60527747600814343379128667689, 4.24584048939253670236584122066, 5.01838900144260653378687226482, 5.82960031217811338004398361736, 6.76411430020496281381540383652, 7.85066129983467539678291709418, 8.967661658105410516488488524905, 9.736882992842253672160300960590, 10.27714359445013524033294975968

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