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Properties

Label 2-950-475.111-c1-0-3
Degree $2$
Conductor $950$
Sign $-0.593 - 0.804i$
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  + (−0.997 − 0.0697i)2-s + (0.0163 + 0.468i)3-s + (0.990 + 0.139i)4-s + (−1.72 + 1.42i)5-s + (0.0163 − 0.468i)6-s + (1.96 + 3.40i)7-s + (−0.978 − 0.207i)8-s + (2.77 − 0.193i)9-s + (1.81 − 1.30i)10-s + (0.484 + 4.60i)11-s + (−0.0490 + 0.466i)12-s + (0.00514 + 0.00762i)13-s + (−1.72 − 3.53i)14-s + (−0.696 − 0.783i)15-s + (0.961 + 0.275i)16-s + (−0.904 + 2.23i)17-s + ⋯
L(s)  = 1  + (−0.705 − 0.0493i)2-s + (0.00944 + 0.270i)3-s + (0.495 + 0.0695i)4-s + (−0.769 + 0.638i)5-s + (0.00667 − 0.191i)6-s + (0.743 + 1.28i)7-s + (−0.345 − 0.0735i)8-s + (0.924 − 0.0646i)9-s + (0.574 − 0.412i)10-s + (0.146 + 1.38i)11-s + (−0.0141 + 0.134i)12-s + (0.00142 + 0.00211i)13-s + (−0.461 − 0.945i)14-s + (−0.179 − 0.202i)15-s + (0.240 + 0.0689i)16-s + (−0.219 + 0.542i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.593 - 0.804i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.593 - 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.593 - 0.804i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.593 - 0.804i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.459453 + 0.909496i\)
\(L(\frac12)\) \(\approx\) \(0.459453 + 0.909496i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.997 + 0.0697i)T \)
5 \( 1 + (1.72 - 1.42i)T \)
19 \( 1 + (0.681 + 4.30i)T \)
good3 \( 1 + (-0.0163 - 0.468i)T + (-2.99 + 0.209i)T^{2} \)
7 \( 1 + (-1.96 - 3.40i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.484 - 4.60i)T + (-10.7 + 2.28i)T^{2} \)
13 \( 1 + (-0.00514 - 0.00762i)T + (-4.86 + 12.0i)T^{2} \)
17 \( 1 + (0.904 - 2.23i)T + (-12.2 - 11.8i)T^{2} \)
23 \( 1 + (-1.98 - 1.91i)T + (0.802 + 22.9i)T^{2} \)
29 \( 1 + (0.886 + 2.19i)T + (-20.8 + 20.1i)T^{2} \)
31 \( 1 + (-1.75 + 1.94i)T + (-3.24 - 30.8i)T^{2} \)
37 \( 1 + (2.58 + 1.88i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (3.43 + 0.984i)T + (34.7 + 21.7i)T^{2} \)
43 \( 1 + (0.892 - 5.06i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-2.66 - 6.60i)T + (-33.8 + 32.6i)T^{2} \)
53 \( 1 + (2.58 + 0.363i)T + (50.9 + 14.6i)T^{2} \)
59 \( 1 + (-2.84 - 11.4i)T + (-52.0 + 27.6i)T^{2} \)
61 \( 1 + (-2.18 - 2.11i)T + (2.12 + 60.9i)T^{2} \)
67 \( 1 + (3.39 + 2.12i)T + (29.3 + 60.2i)T^{2} \)
71 \( 1 + (8.28 + 4.40i)T + (39.7 + 58.8i)T^{2} \)
73 \( 1 + (-3.87 + 5.74i)T + (-27.3 - 67.6i)T^{2} \)
79 \( 1 + (0.462 + 13.2i)T + (-78.8 + 5.51i)T^{2} \)
83 \( 1 + (3.61 - 4.01i)T + (-8.67 - 82.5i)T^{2} \)
89 \( 1 + (-5.91 + 1.69i)T + (75.4 - 47.1i)T^{2} \)
97 \( 1 + (16.0 - 10.0i)T + (42.5 - 87.1i)T^{2} \)
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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

−10.31330719916285477806965489270, −9.432730017099377161036152196833, −8.779054494845022516544750830578, −7.77677542196884084688870514728, −7.19685861981254716628161801814, −6.29295594231629705210985110651, −4.93726710029165514694880696431, −4.15076084121811769698603913917, −2.72671669718598891744807548061, −1.73290597511133659542355543835, 0.65122006428704419107552866846, 1.53062231589302521088679732809, 3.45443927046761481352638609245, 4.30379502650752999969779026266, 5.33758464363898706381122523351, 6.75462405057606991416957213667, 7.31758936735548301933511825540, 8.220188944133625911934115398785, 8.567323530327498911783959808770, 9.793285332467932318135350779648

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