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Properties

Label 2-950-95.4-c1-0-19
Degree $2$
Conductor $950$
Sign $0.862 - 0.505i$
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  + (−0.342 + 0.939i)2-s + (2.49 − 0.439i)3-s + (−0.766 − 0.642i)4-s + (−0.439 + 2.49i)6-s + (−0.565 − 0.326i)7-s + (0.866 − 0.500i)8-s + (3.20 − 1.16i)9-s + (0.5 + 0.866i)11-s + (−2.19 − 1.26i)12-s + (2.83 + 0.5i)13-s + (0.5 − 0.419i)14-s + (0.173 + 0.984i)16-s + (−0.160 + 0.439i)17-s + 3.41i·18-s + (4.07 + 1.55i)19-s + ⋯
L(s)  = 1  + (−0.241 + 0.664i)2-s + (1.43 − 0.253i)3-s + (−0.383 − 0.321i)4-s + (−0.179 + 1.01i)6-s + (−0.213 − 0.123i)7-s + (0.306 − 0.176i)8-s + (1.06 − 0.388i)9-s + (0.150 + 0.261i)11-s + (−0.633 − 0.365i)12-s + (0.786 + 0.138i)13-s + (0.133 − 0.112i)14-s + (0.0434 + 0.246i)16-s + (−0.0388 + 0.106i)17-s + 0.804i·18-s + (0.934 + 0.355i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.862 - 0.505i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.862 - 0.505i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.862 - 0.505i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.862 - 0.505i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.20578 + 0.598702i\)
\(L(\frac12)\) \(\approx\) \(2.20578 + 0.598702i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.342 - 0.939i)T \)
5 \( 1 \)
19 \( 1 + (-4.07 - 1.55i)T \)
good3 \( 1 + (-2.49 + 0.439i)T + (2.81 - 1.02i)T^{2} \)
7 \( 1 + (0.565 + 0.326i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.83 - 0.5i)T + (12.2 + 4.44i)T^{2} \)
17 \( 1 + (0.160 - 0.439i)T + (-13.0 - 10.9i)T^{2} \)
23 \( 1 + (-2.15 + 2.56i)T + (-3.99 - 22.6i)T^{2} \)
29 \( 1 + (-6.41 + 2.33i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-2.03 + 3.51i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 7.63iT - 37T^{2} \)
41 \( 1 + (0.854 + 4.84i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (-2.02 - 2.40i)T + (-7.46 + 42.3i)T^{2} \)
47 \( 1 + (-2.19 - 6.02i)T + (-36.0 + 30.2i)T^{2} \)
53 \( 1 + (2.48 - 2.96i)T + (-9.20 - 52.1i)T^{2} \)
59 \( 1 + (5.68 + 2.06i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (3.96 + 3.32i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (0.300 + 0.826i)T + (-51.3 + 43.0i)T^{2} \)
71 \( 1 + (-9.35 + 7.84i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (0.502 - 0.0885i)T + (68.5 - 24.9i)T^{2} \)
79 \( 1 + (-1.51 - 8.58i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (15.5 + 8.95i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-3.06 + 17.3i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (4.96 - 13.6i)T + (-74.3 - 62.3i)T^{2} \)
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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

−9.716726604100987888006059106783, −9.145990987082061377294691986902, −8.293627946889663854370597895527, −7.83372963751159063822477846167, −6.87195248281098488775930319499, −6.10338033089116145612832657691, −4.75268225915393069805631876197, −3.70336836051957133766459706157, −2.73309345598229255846344000053, −1.32727513694209879325053481170, 1.31156216178423789928539899250, 2.72359486620732901126466573544, 3.29319814708016049781274895045, 4.21830912419578165206704901488, 5.45007235083940824024273694496, 6.81214319760649907131154632231, 7.81077649861131288083027144141, 8.553182788669854591793277896172, 9.137539851654080781124305747618, 9.760080564605378455965021645224

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