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Properties

Label 2-950-95.64-c1-0-23
Degree $2$
Conductor $950$
Sign $-0.967 + 0.252i$
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  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s − 4i·7-s + 0.999i·8-s + (−1.5 + 2.59i)9-s − 11-s + (−1.73 − i)13-s + (2 + 3.46i)14-s + (−0.5 − 0.866i)16-s + (2.59 − 1.5i)17-s − 3i·18-s + (−4 + 1.73i)19-s + (0.866 − 0.5i)22-s + (−3.46 − 2i)23-s + 1.99·26-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s − 1.51i·7-s + 0.353i·8-s + (−0.5 + 0.866i)9-s − 0.301·11-s + (−0.480 − 0.277i)13-s + (0.534 + 0.925i)14-s + (−0.125 − 0.216i)16-s + (0.630 − 0.363i)17-s − 0.707i·18-s + (−0.917 + 0.397i)19-s + (0.184 − 0.106i)22-s + (−0.722 − 0.417i)23-s + 0.392·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 + 0.252i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.967 + 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.967 + 0.252i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.967 + 0.252i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0164301 - 0.127791i\)
\(L(\frac12)\) \(\approx\) \(0.0164301 - 0.127791i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 \)
19 \( 1 + (4 - 1.73i)T \)
good3 \( 1 + (1.5 - 2.59i)T^{2} \)
7 \( 1 + 4iT - 7T^{2} \)
11 \( 1 + T + 11T^{2} \)
13 \( 1 + (1.73 + i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.59 + 1.5i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (3.46 + 2i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3 - 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 + (-1.5 - 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (10.3 - 6i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.19 + 3i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.46 - 2i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.5 - 7.79i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6 + 10.3i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (12.9 + 7.5i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (3 + 5.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (8.66 - 5i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (7 + 12.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 4iT - 83T^{2} \)
89 \( 1 + (0.5 - 0.866i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.866 - 0.5i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.884756222371515592987752759087, −8.637468438791074791122456586267, −7.87108633141805306467972112427, −7.36654645058569213024308828415, −6.41234531765191042081676305317, −5.31862464841609411557547049804, −4.43576366485514292525272644486, −3.13530812597439691464877968150, −1.69702765497134324801395264333, −0.06902544753495799583094490710, 1.94558489661528996160772268840, 2.83121471976045221629937198561, 3.97374035015413308731977684473, 5.43789643344384683988432976920, 6.07938407972285451453570621602, 7.12654592791856077799503944032, 8.283098497029460116337596713756, 8.754976910318186409084274281204, 9.554336322584050999997393199040, 10.20762934458527114800979644983

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