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Properties

Label 2-950-25.6-c1-0-23
Degree $2$
Conductor $950$
Sign $-0.594 + 0.803i$
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  + (−0.309 − 0.951i)2-s + (−0.628 + 0.456i)3-s + (−0.809 + 0.587i)4-s + (−0.766 + 2.10i)5-s + (0.628 + 0.456i)6-s − 4.62·7-s + (0.809 + 0.587i)8-s + (−0.740 + 2.27i)9-s + (2.23 + 0.0798i)10-s + (0.182 + 0.561i)11-s + (0.240 − 0.738i)12-s + (1.06 − 3.28i)13-s + (1.43 + 4.40i)14-s + (−0.477 − 1.66i)15-s + (0.309 − 0.951i)16-s + (6.08 + 4.42i)17-s + ⋯
L(s)  = 1  + (−0.218 − 0.672i)2-s + (−0.362 + 0.263i)3-s + (−0.404 + 0.293i)4-s + (−0.342 + 0.939i)5-s + (0.256 + 0.186i)6-s − 1.74·7-s + (0.286 + 0.207i)8-s + (−0.246 + 0.759i)9-s + (0.706 + 0.0252i)10-s + (0.0549 + 0.169i)11-s + (0.0692 − 0.213i)12-s + (0.295 − 0.909i)13-s + (0.382 + 1.17i)14-s + (−0.123 − 0.431i)15-s + (0.0772 − 0.237i)16-s + (1.47 + 1.07i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.594 + 0.803i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.594 + 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.594 + 0.803i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (381, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.594 + 0.803i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.114252 - 0.226628i\)
\(L(\frac12)\) \(\approx\) \(0.114252 - 0.226628i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.309 + 0.951i)T \)
5 \( 1 + (0.766 - 2.10i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
good3 \( 1 + (0.628 - 0.456i)T + (0.927 - 2.85i)T^{2} \)
7 \( 1 + 4.62T + 7T^{2} \)
11 \( 1 + (-0.182 - 0.561i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-1.06 + 3.28i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-6.08 - 4.42i)T + (5.25 + 16.1i)T^{2} \)
23 \( 1 + (2.36 + 7.27i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (7.65 - 5.55i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (7.03 + 5.10i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-1.23 + 3.79i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.136 - 0.420i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 7.68T + 43T^{2} \)
47 \( 1 + (-7.46 + 5.42i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (2.55 - 1.85i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-2.02 + 6.22i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (2.74 + 8.44i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (0.436 + 0.317i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (4.90 - 3.56i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (4.05 + 12.4i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-4.34 + 3.15i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-0.322 - 0.234i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (-2.28 - 7.01i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (8.76 - 6.37i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.03849248664231863993363228287, −9.218827929698535698783790061688, −8.012422534439668776957498899371, −7.35281752749161825627178944965, −6.13712584915996422735730266940, −5.58311473315298781930771383373, −3.89636736962705738056325928772, −3.37663557423372144377312141365, −2.33565121440550195827689936795, −0.15692481495434240077082716977, 1.10105178096932174998532524857, 3.29658220068387570628795748386, 4.05392985814210471140222380070, 5.69913558044471251657657094329, 5.78923581049901969404018377072, 7.08107581947751785905908555260, 7.52125104140008705700770776610, 8.911216475036255244567158952294, 9.403054435841203387548353272038, 9.808125595967506342871749965703

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