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Properties

Label 2-950-95.27-c1-0-27
Degree $2$
Conductor $950$
Sign $0.984 + 0.175i$
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  + (0.258 + 0.965i)2-s + (3.24 − 0.869i)3-s + (−0.866 + 0.499i)4-s + (1.67 + 2.90i)6-s + (−2.68 − 2.68i)7-s + (−0.707 − 0.707i)8-s + (7.16 − 4.13i)9-s + 1.84·11-s + (−2.37 + 2.37i)12-s + (0.593 − 2.21i)13-s + (1.90 − 3.29i)14-s + (0.500 − 0.866i)16-s + (−0.212 + 0.0569i)17-s + (5.85 + 5.85i)18-s + (3.81 + 2.11i)19-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (1.87 − 0.501i)3-s + (−0.433 + 0.249i)4-s + (0.685 + 1.18i)6-s + (−1.01 − 1.01i)7-s + (−0.249 − 0.249i)8-s + (2.38 − 1.37i)9-s + 0.555·11-s + (−0.685 + 0.685i)12-s + (0.164 − 0.613i)13-s + (0.508 − 0.880i)14-s + (0.125 − 0.216i)16-s + (−0.0515 + 0.0138i)17-s + (1.37 + 1.37i)18-s + (0.874 + 0.484i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.175i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 + 0.175i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.984 + 0.175i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (407, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.984 + 0.175i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.96820 - 0.262303i\)
\(L(\frac12)\) \(\approx\) \(2.96820 - 0.262303i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 - 0.965i)T \)
5 \( 1 \)
19 \( 1 + (-3.81 - 2.11i)T \)
good3 \( 1 + (-3.24 + 0.869i)T + (2.59 - 1.5i)T^{2} \)
7 \( 1 + (2.68 + 2.68i)T + 7iT^{2} \)
11 \( 1 - 1.84T + 11T^{2} \)
13 \( 1 + (-0.593 + 2.21i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (0.212 - 0.0569i)T + (14.7 - 8.5i)T^{2} \)
23 \( 1 + (3.25 + 0.872i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (1.14 + 1.97i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 3.03iT - 31T^{2} \)
37 \( 1 + (5.08 - 5.08i)T - 37iT^{2} \)
41 \( 1 + (-8.86 - 5.11i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.643 - 2.40i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (-0.218 + 0.816i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (0.323 - 1.20i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (3.04 - 5.27i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.23 - 5.60i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.08 + 1.63i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (2.74 + 1.58i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-0.572 - 2.13i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (2.94 - 5.10i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.543 - 0.543i)T - 83iT^{2} \)
89 \( 1 + (8.67 + 15.0i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.62 - 6.07i)T + (-84.0 + 48.5i)T^{2} \)
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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

−9.753109346196432883182014715348, −9.050956200720203747026842761902, −8.177098472208269511690848785689, −7.52295604314561952270874196467, −6.90394206398597240398949988203, −6.06258946642055136861304718206, −4.32494998230821288544221133226, −3.59332650480788422404924611808, −2.88910385347750470925162702771, −1.21058031021100926620243391877, 1.86812918263669183335438177213, 2.72263824316217739023694461825, 3.52856901173757744250837977999, 4.24238386248892303110550921364, 5.52258449189329679999046543215, 6.80551849430604911984434770020, 7.81690277375738350676873813676, 8.906372436620980142033766842226, 9.210093743646498118497265251520, 9.705630518112312322400146770458

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