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Properties

Label 2-950-95.9-c1-0-0
Degree $2$
Conductor $950$
Sign $-0.446 - 0.894i$
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 + (−0.536 − 0.639i)3-s + (0.939 − 0.342i)4-s + (0.639 + 0.536i)6-s + (−3.62 − 2.09i)7-s + (−0.866 + 0.5i)8-s + (0.399 − 2.26i)9-s + (−2.69 − 4.66i)11-s + (−0.723 − 0.417i)12-s + (−2.52 + 3.01i)13-s + (3.93 + 1.43i)14-s + (0.766 − 0.642i)16-s + (2.10 − 0.371i)17-s + 2.30i·18-s + (2.92 + 3.22i)19-s + ⋯
L(s)  = 1  + (−0.696 + 0.122i)2-s + (−0.309 − 0.369i)3-s + (0.469 − 0.171i)4-s + (0.261 + 0.219i)6-s + (−1.36 − 0.790i)7-s + (−0.306 + 0.176i)8-s + (0.133 − 0.755i)9-s + (−0.812 − 1.40i)11-s + (−0.208 − 0.120i)12-s + (−0.700 + 0.835i)13-s + (1.05 + 0.382i)14-s + (0.191 − 0.160i)16-s + (0.511 − 0.0901i)17-s + 0.542i·18-s + (0.671 + 0.740i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.446 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.446 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.446 - 0.894i$
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: \((2,\ 950,\ (\ :1/2),\ -0.446 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0397435 + 0.0642574i\)
\(L(\frac12)\) \(\approx\) \(0.0397435 + 0.0642574i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.92 - 3.22i)T \)
good3 \( 1 + (0.536 + 0.639i)T + (-0.520 + 2.95i)T^{2} \)
7 \( 1 + (3.62 + 2.09i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.69 + 4.66i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.52 - 3.01i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 + (-2.10 + 0.371i)T + (15.9 - 5.81i)T^{2} \)
23 \( 1 + (-0.976 - 2.68i)T + (-17.6 + 14.7i)T^{2} \)
29 \( 1 + (1.34 - 7.65i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (2.86 - 4.96i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 1.84iT - 37T^{2} \)
41 \( 1 + (-4.67 + 3.92i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-4.31 + 11.8i)T + (-32.9 - 27.6i)T^{2} \)
47 \( 1 + (4.31 + 0.760i)T + (44.1 + 16.0i)T^{2} \)
53 \( 1 + (-4.44 - 12.2i)T + (-40.6 + 34.0i)T^{2} \)
59 \( 1 + (-1.36 - 7.75i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (10.3 - 3.74i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (10.3 + 1.82i)T + (62.9 + 22.9i)T^{2} \)
71 \( 1 + (12.6 + 4.61i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (-2.16 - 2.58i)T + (-12.6 + 71.8i)T^{2} \)
79 \( 1 + (-3.96 + 3.32i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-7.56 - 4.36i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.52 - 2.12i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (-0.851 + 0.150i)T + (91.1 - 33.1i)T^{2} \)
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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

−10.36587813349765079777090987436, −9.331102439716896966415114943085, −8.924134138475445007217128949666, −7.40368708772511605158889264149, −7.22130253094073106869175936729, −6.13717861726278453825536446242, −5.50108676250173816525247054218, −3.72686725599960169474472290321, −3.02243527416888673955483387404, −1.16846284720526614054300332861, 0.05002892902451528923455754102, 2.29423628400814920891436643708, 2.96686426246210532569232799263, 4.56632958708426722489732338230, 5.45165795093886171648391210653, 6.36092039910249471716744597014, 7.51306577564588407279856779562, 7.923625336424253463003211527730, 9.355981384389227621212494515120, 9.793174482766853673417314176647

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