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Properties

Label 2-950-95.78-c1-0-4
Degree $2$
Conductor $950$
Sign $-0.574 - 0.818i$
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  + (−0.819 + 0.573i)2-s + (1.52 − 0.133i)3-s + (0.342 − 0.939i)4-s + (−1.17 + 0.984i)6-s + (−0.848 − 0.227i)7-s + (0.258 + 0.965i)8-s + (−0.642 + 0.113i)9-s + (−1.39 + 2.40i)11-s + (0.396 − 1.47i)12-s + (−0.512 + 5.85i)13-s + (0.825 − 0.300i)14-s + (−0.766 − 0.642i)16-s + (−0.306 − 0.437i)17-s + (0.461 − 0.461i)18-s + (−3.66 − 2.35i)19-s + ⋯
L(s)  = 1  + (−0.579 + 0.405i)2-s + (0.881 − 0.0770i)3-s + (0.171 − 0.469i)4-s + (−0.479 + 0.402i)6-s + (−0.320 − 0.0859i)7-s + (0.0915 + 0.341i)8-s + (−0.214 + 0.0377i)9-s + (−0.419 + 0.726i)11-s + (0.114 − 0.427i)12-s + (−0.142 + 1.62i)13-s + (0.220 − 0.0802i)14-s + (−0.191 − 0.160i)16-s + (−0.0742 − 0.106i)17-s + (0.108 − 0.108i)18-s + (−0.841 − 0.539i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.574 - 0.818i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.574 - 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.574 - 0.818i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (743, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.574 - 0.818i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.455763 + 0.876369i\)
\(L(\frac12)\) \(\approx\) \(0.455763 + 0.876369i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.819 - 0.573i)T \)
5 \( 1 \)
19 \( 1 + (3.66 + 2.35i)T \)
good3 \( 1 + (-1.52 + 0.133i)T + (2.95 - 0.520i)T^{2} \)
7 \( 1 + (0.848 + 0.227i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (1.39 - 2.40i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.512 - 5.85i)T + (-12.8 - 2.25i)T^{2} \)
17 \( 1 + (0.306 + 0.437i)T + (-5.81 + 15.9i)T^{2} \)
23 \( 1 + (-2.73 - 5.86i)T + (-14.7 + 17.6i)T^{2} \)
29 \( 1 + (-0.259 - 1.47i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (3.95 - 2.28i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.22 - 4.22i)T + 37iT^{2} \)
41 \( 1 + (-2.19 + 2.61i)T + (-7.11 - 40.3i)T^{2} \)
43 \( 1 + (-6.64 - 3.10i)T + (27.6 + 32.9i)T^{2} \)
47 \( 1 + (-1.49 - 1.04i)T + (16.0 + 44.1i)T^{2} \)
53 \( 1 + (-3.01 + 1.40i)T + (34.0 - 40.6i)T^{2} \)
59 \( 1 + (1.12 - 6.39i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (0.643 + 0.234i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (0.550 - 0.786i)T + (-22.9 - 62.9i)T^{2} \)
71 \( 1 + (1.03 + 2.83i)T + (-54.3 + 45.6i)T^{2} \)
73 \( 1 + (-0.840 - 9.60i)T + (-71.8 + 12.6i)T^{2} \)
79 \( 1 + (-1.70 - 1.42i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-3.41 + 12.7i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (-10.4 + 8.78i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (5.53 - 3.87i)T + (33.1 - 91.1i)T^{2} \)
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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

−10.02088651365371482953879106420, −9.127149129514030390892198501132, −8.955228467042222531329545319548, −7.76798482340583586455829041959, −7.16989070460188500424269877408, −6.33780405627979739787340983882, −5.12152451425357579387037381355, −4.05931505293110692744918933644, −2.73721611550066099230498013057, −1.77368456951081687122987996905, 0.47103186982575073714160656384, 2.38976912930646388337546565242, 3.02650827071747276558572682195, 3.98704046964620676010231193148, 5.48704180965820436219144916960, 6.35087990933525977346351078645, 7.74450249733675351249213505346, 8.137870492591849553831217331268, 8.908311442061450067809813272774, 9.626788355482057118576108320029

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