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Properties

Label 2-950-95.24-c1-0-15
Degree $2$
Conductor $950$
Sign $-0.997 - 0.0641i$
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  + (−0.342 − 0.939i)2-s + (−2.84 − 0.501i)3-s + (−0.766 + 0.642i)4-s + (0.501 + 2.84i)6-s + (−3.49 + 2.01i)7-s + (0.866 + 0.500i)8-s + (5.02 + 1.82i)9-s + (2.11 − 3.65i)11-s + (2.50 − 1.44i)12-s + (5.44 − 0.959i)13-s + (3.09 + 2.59i)14-s + (0.173 − 0.984i)16-s + (0.523 + 1.43i)17-s − 5.34i·18-s + (0.805 + 4.28i)19-s + ⋯
L(s)  = 1  + (−0.241 − 0.664i)2-s + (−1.64 − 0.289i)3-s + (−0.383 + 0.321i)4-s + (0.204 + 1.16i)6-s + (−1.32 + 0.762i)7-s + (0.306 + 0.176i)8-s + (1.67 + 0.609i)9-s + (0.636 − 1.10i)11-s + (0.722 − 0.417i)12-s + (1.50 − 0.266i)13-s + (0.826 + 0.693i)14-s + (0.0434 − 0.246i)16-s + (0.127 + 0.349i)17-s − 1.26i·18-s + (0.184 + 0.982i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0641i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0641i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.997 - 0.0641i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.997 - 0.0641i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00784047 + 0.244290i\)
\(L(\frac12)\) \(\approx\) \(0.00784047 + 0.244290i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.342 + 0.939i)T \)
5 \( 1 \)
19 \( 1 + (-0.805 - 4.28i)T \)
good3 \( 1 + (2.84 + 0.501i)T + (2.81 + 1.02i)T^{2} \)
7 \( 1 + (3.49 - 2.01i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.11 + 3.65i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-5.44 + 0.959i)T + (12.2 - 4.44i)T^{2} \)
17 \( 1 + (-0.523 - 1.43i)T + (-13.0 + 10.9i)T^{2} \)
23 \( 1 + (3.83 + 4.57i)T + (-3.99 + 22.6i)T^{2} \)
29 \( 1 + (4.97 + 1.81i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (1.35 + 2.35i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 8.71iT - 37T^{2} \)
41 \( 1 + (0.560 - 3.17i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (-1.46 + 1.74i)T + (-7.46 - 42.3i)T^{2} \)
47 \( 1 + (-3.61 + 9.93i)T + (-36.0 - 30.2i)T^{2} \)
53 \( 1 + (1.66 + 1.97i)T + (-9.20 + 52.1i)T^{2} \)
59 \( 1 + (9.90 - 3.60i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (3.45 - 2.89i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-3.76 + 10.3i)T + (-51.3 - 43.0i)T^{2} \)
71 \( 1 + (5.73 + 4.81i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (1.94 + 0.342i)T + (68.5 + 24.9i)T^{2} \)
79 \( 1 + (2.15 - 12.2i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (3.32 - 1.91i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.96 + 11.1i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (4.66 + 12.8i)T + (-74.3 + 62.3i)T^{2} \)
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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

−9.906455880385027984140833227968, −8.911842338776998409016180370135, −8.107672347656163294430303869567, −6.63226635407802761971092613811, −6.01420887838508982347309754391, −5.67854051550449398334528165279, −4.07425055578694227414571598961, −3.22076183284415297354858017047, −1.47629757012266296835290783821, −0.18678423798057961406064780760, 1.20908499399583428480629249493, 3.70575448346870798964044900789, 4.39363782414470389800397386490, 5.56161216828731051804208558146, 6.22885724618067528966641769792, 6.89964842433086117648695366587, 7.47745724005581059779071221837, 9.250210062707699146162553776316, 9.495938273088293192091908460200, 10.54178243284241313121291929994

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