Properties

Label 2-950-95.54-c1-0-0
Degree $2$
Conductor $950$
Sign $0.644 - 0.764i$
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  + (0.642 − 0.766i)2-s + (−0.931 − 2.55i)3-s + (−0.173 − 0.984i)4-s + (−2.55 − 0.931i)6-s + (−4.04 + 2.33i)7-s + (−0.866 − 0.500i)8-s + (−3.37 + 2.83i)9-s + (0.0690 − 0.119i)11-s + (−2.35 + 1.36i)12-s + (−1.30 + 3.59i)13-s + (−0.810 + 4.59i)14-s + (−0.939 + 0.342i)16-s + (3.80 − 4.53i)17-s + 4.41i·18-s + (3.96 + 1.80i)19-s + ⋯
L(s)  = 1  + (0.454 − 0.541i)2-s + (−0.537 − 1.47i)3-s + (−0.0868 − 0.492i)4-s + (−1.04 − 0.380i)6-s + (−1.52 + 0.882i)7-s + (−0.306 − 0.176i)8-s + (−1.12 + 0.945i)9-s + (0.0208 − 0.0360i)11-s + (−0.680 + 0.392i)12-s + (−0.362 + 0.996i)13-s + (−0.216 + 1.22i)14-s + (−0.234 + 0.0855i)16-s + (0.922 − 1.09i)17-s + 1.03i·18-s + (0.910 + 0.413i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.644 - 0.764i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.644 - 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.644 - 0.764i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.644 - 0.764i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.262512 + 0.121992i\)
\(L(\frac12)\) \(\approx\) \(0.262512 + 0.121992i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.642 + 0.766i)T \)
5 \( 1 \)
19 \( 1 + (-3.96 - 1.80i)T \)
good3 \( 1 + (0.931 + 2.55i)T + (-2.29 + 1.92i)T^{2} \)
7 \( 1 + (4.04 - 2.33i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.0690 + 0.119i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.30 - 3.59i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + (-3.80 + 4.53i)T + (-2.95 - 16.7i)T^{2} \)
23 \( 1 + (5.93 - 1.04i)T + (21.6 - 7.86i)T^{2} \)
29 \( 1 + (4.90 - 4.11i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (-2.80 - 4.86i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 2.34iT - 37T^{2} \)
41 \( 1 + (1.71 - 0.622i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (6.49 + 1.14i)T + (40.4 + 14.7i)T^{2} \)
47 \( 1 + (4.52 + 5.39i)T + (-8.16 + 46.2i)T^{2} \)
53 \( 1 + (-3.64 + 0.642i)T + (49.8 - 18.1i)T^{2} \)
59 \( 1 + (6.81 + 5.71i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-1.58 - 9.00i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-2.81 - 3.35i)T + (-11.6 + 65.9i)T^{2} \)
71 \( 1 + (-0.856 + 4.85i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (-3.43 - 9.44i)T + (-55.9 + 46.9i)T^{2} \)
79 \( 1 + (13.0 - 4.76i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (10.3 - 5.98i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (11.7 + 4.28i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (4.35 - 5.18i)T + (-16.8 - 95.5i)T^{2} \)
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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

−9.997498053888768356606605486228, −9.594724942145501335974122058312, −8.489341416257618526449914754540, −7.23814716157992379151068948937, −6.72876200817577118298504057915, −5.85323434521836520013612026479, −5.23689425962454045754249329702, −3.53257015772398068399173537648, −2.61875794951213816827931494121, −1.48664020865748834216701480381, 0.12305046822119383224964554418, 3.14437774457254171933119483374, 3.73624373352856541266849362336, 4.53911413817731311750559096569, 5.71714857906302435460752149169, 6.08315330565230320956139640134, 7.29774085519062605214001741656, 8.148546889841378171928708942536, 9.540640110090994078320515267042, 9.893665215052344143066338025068

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