Properties

Label 2-950-95.9-c1-0-10
Degree $2$
Conductor $950$
Sign $0.131 - 0.991i$
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  + (0.984 − 0.173i)2-s + (2.09 + 2.49i)3-s + (0.939 − 0.342i)4-s + (2.49 + 2.09i)6-s + (−0.964 − 0.556i)7-s + (0.866 − 0.5i)8-s + (−1.31 + 7.46i)9-s + (0.0761 + 0.131i)11-s + (2.81 + 1.62i)12-s + (−2.23 + 2.66i)13-s + (−1.04 − 0.380i)14-s + (0.766 − 0.642i)16-s + (0.164 − 0.0290i)17-s + 7.58i·18-s + (3.08 − 3.08i)19-s + ⋯
L(s)  = 1  + (0.696 − 0.122i)2-s + (1.20 + 1.43i)3-s + (0.469 − 0.171i)4-s + (1.01 + 0.853i)6-s + (−0.364 − 0.210i)7-s + (0.306 − 0.176i)8-s + (−0.438 + 2.48i)9-s + (0.0229 + 0.0397i)11-s + (0.813 + 0.469i)12-s + (−0.619 + 0.738i)13-s + (−0.279 − 0.101i)14-s + (0.191 − 0.160i)16-s + (0.0399 − 0.00704i)17-s + 1.78i·18-s + (0.707 − 0.706i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.131 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.131 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.131 - 0.991i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.131 - 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.53497 + 2.22147i\)
\(L(\frac12)\) \(\approx\) \(2.53497 + 2.22147i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.984 + 0.173i)T \)
5 \( 1 \)
19 \( 1 + (-3.08 + 3.08i)T \)
good3 \( 1 + (-2.09 - 2.49i)T + (-0.520 + 2.95i)T^{2} \)
7 \( 1 + (0.964 + 0.556i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.0761 - 0.131i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.23 - 2.66i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 + (-0.164 + 0.0290i)T + (15.9 - 5.81i)T^{2} \)
23 \( 1 + (-2.12 - 5.83i)T + (-17.6 + 14.7i)T^{2} \)
29 \( 1 + (0.881 - 4.99i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-4.34 + 7.52i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 3.71iT - 37T^{2} \)
41 \( 1 + (-3.70 + 3.10i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-3.46 + 9.50i)T + (-32.9 - 27.6i)T^{2} \)
47 \( 1 + (-8.28 - 1.46i)T + (44.1 + 16.0i)T^{2} \)
53 \( 1 + (4.87 + 13.3i)T + (-40.6 + 34.0i)T^{2} \)
59 \( 1 + (2.35 + 13.3i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (4.92 - 1.79i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (2.55 + 0.450i)T + (62.9 + 22.9i)T^{2} \)
71 \( 1 + (-1.70 - 0.622i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (-6.53 - 7.79i)T + (-12.6 + 71.8i)T^{2} \)
79 \( 1 + (4.00 - 3.35i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (5.26 + 3.03i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (9.18 + 7.70i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (-12.2 + 2.16i)T + (91.1 - 33.1i)T^{2} \)
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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

−9.986570463578471107855843345078, −9.559327216279030564945097472894, −8.836515236117540305681230407304, −7.73362742131097128660414858999, −6.94492656454487225480433224250, −5.46314591339375661634227374898, −4.74338325672724814963020019661, −3.86968454413483178716971466757, −3.15524277266027806234880600478, −2.16766402697596425136593033533, 1.18960397562885687475822385884, 2.65717137492988561687550312356, 3.00752956834814506747833407985, 4.35329699483095333070164248289, 5.81297574065222116787677998909, 6.46247111756481211548223323341, 7.45758025606781907911454683163, 7.86334274640162157760964971193, 8.806239566377280033238839597773, 9.627498515517130536367271950173

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