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Properties

Label 2-950-25.16-c1-0-18
Degree $2$
Conductor $950$
Sign $0.597 - 0.802i$
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  + (−0.809 − 0.587i)2-s + (0.445 + 1.36i)3-s + (0.309 + 0.951i)4-s + (−0.223 + 2.22i)5-s + (0.445 − 1.36i)6-s − 0.156·7-s + (0.309 − 0.951i)8-s + (0.748 − 0.543i)9-s + (1.48 − 1.66i)10-s + (4.02 + 2.92i)11-s + (−1.16 + 0.846i)12-s + (4.49 − 3.26i)13-s + (0.126 + 0.0922i)14-s + (−3.14 + 0.684i)15-s + (−0.809 + 0.587i)16-s + (1.74 − 5.37i)17-s + ⋯
L(s)  = 1  + (−0.572 − 0.415i)2-s + (0.256 + 0.790i)3-s + (0.154 + 0.475i)4-s + (−0.0999 + 0.994i)5-s + (0.181 − 0.559i)6-s − 0.0592·7-s + (0.109 − 0.336i)8-s + (0.249 − 0.181i)9-s + (0.470 − 0.527i)10-s + (1.21 + 0.880i)11-s + (−0.336 + 0.244i)12-s + (1.24 − 0.906i)13-s + (0.0339 + 0.0246i)14-s + (−0.812 + 0.176i)15-s + (−0.202 + 0.146i)16-s + (0.423 − 1.30i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.597 - 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.597 - 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.597 - 0.802i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.597 - 0.802i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35561 + 0.680737i\)
\(L(\frac12)\) \(\approx\) \(1.35561 + 0.680737i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 + 0.587i)T \)
5 \( 1 + (0.223 - 2.22i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
good3 \( 1 + (-0.445 - 1.36i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 + 0.156T + 7T^{2} \)
11 \( 1 + (-4.02 - 2.92i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-4.49 + 3.26i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.74 + 5.37i)T + (-13.7 - 9.99i)T^{2} \)
23 \( 1 + (-3.45 - 2.50i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (0.0582 + 0.179i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (0.452 - 1.39i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (2.02 - 1.47i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (1.85 - 1.34i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 9.77T + 43T^{2} \)
47 \( 1 + (3.11 + 9.57i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-3.95 - 12.1i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (11.3 - 8.27i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (8.45 + 6.14i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (2.83 - 8.73i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (-1.39 - 4.30i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (4.86 + 3.53i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (4.03 + 12.4i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-1.28 + 3.95i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (1.45 + 1.05i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-2.08 - 6.43i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.18898187746460485567795590576, −9.392764724171446590462028822439, −8.879925405359259922743332379583, −7.59718166670229020242619232134, −6.99442096440412001494060854668, −6.00441987514943473122651317145, −4.55794679807738627855860693874, −3.58034419388123890623612572823, −2.99697457153156548515484511691, −1.34615666641579814445883696688, 1.07076835867708606875260784279, 1.72996925808117957447211254560, 3.64222786281015605182028132837, 4.61829224527577332659455710234, 6.05217128483666696909430274591, 6.43338301692826953999580078483, 7.53488079520765095315666107176, 8.427564868249604047042304004277, 8.779598174243701752634674795521, 9.558439313456499863791656334830

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