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Properties

Label 2-950-25.11-c1-0-21
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: $\chi_{950} (761, \cdot )$
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

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

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