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Properties

Label 2-950-19.16-c1-0-29
Degree $2$
Conductor $950$
Sign $0.239 + 0.970i$
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  + (0.766 + 0.642i)2-s + (0.541 − 0.197i)3-s + (0.173 + 0.984i)4-s + (0.541 + 0.197i)6-s + (−2.43 − 4.21i)7-s + (−0.500 + 0.866i)8-s + (−2.04 + 1.71i)9-s + (2.68 − 4.64i)11-s + (0.288 + 0.499i)12-s + (−3.62 − 1.31i)13-s + (0.844 − 4.79i)14-s + (−0.939 + 0.342i)16-s + (−1.07 − 0.901i)17-s − 2.66·18-s + (4.35 − 0.226i)19-s + ⋯
L(s)  = 1  + (0.541 + 0.454i)2-s + (0.312 − 0.113i)3-s + (0.0868 + 0.492i)4-s + (0.221 + 0.0804i)6-s + (−0.919 − 1.59i)7-s + (−0.176 + 0.306i)8-s + (−0.681 + 0.571i)9-s + (0.809 − 1.40i)11-s + (0.0831 + 0.144i)12-s + (−1.00 − 0.365i)13-s + (0.225 − 1.28i)14-s + (−0.234 + 0.0855i)16-s + (−0.260 − 0.218i)17-s − 0.628·18-s + (0.998 − 0.0520i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.239 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.239 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.239 + 0.970i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.239 + 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22628 - 0.960627i\)
\(L(\frac12)\) \(\approx\) \(1.22628 - 0.960627i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.766 - 0.642i)T \)
5 \( 1 \)
19 \( 1 + (-4.35 + 0.226i)T \)
good3 \( 1 + (-0.541 + 0.197i)T + (2.29 - 1.92i)T^{2} \)
7 \( 1 + (2.43 + 4.21i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.68 + 4.64i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.62 + 1.31i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (1.07 + 0.901i)T + (2.95 + 16.7i)T^{2} \)
23 \( 1 + (0.927 + 5.25i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (2.78 - 2.33i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (4.10 + 7.10i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 10.4T + 37T^{2} \)
41 \( 1 + (-1.79 + 0.652i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (0.256 - 1.45i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-2.39 + 2.00i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + (-0.312 - 1.77i)T + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (5.61 + 4.70i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-1.40 - 7.99i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (6.46 - 5.42i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-1.04 + 5.93i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (1.19 - 0.436i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (-15.6 + 5.68i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (-7.23 - 12.5i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (9.02 + 3.28i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (-4.65 - 3.90i)T + (16.8 + 95.5i)T^{2} \)
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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

−9.767386146602822157894364515014, −9.002527123656504121709948060366, −7.86795769784030066164812934522, −7.38359747474681424133216658660, −6.42578374572299182740459508093, −5.64957517580445595915331980291, −4.43031932396788010917346464639, −3.53400173432318061953268551716, −2.72308554553931861674381073167, −0.55726159450298906119556662364, 1.94939022457793936775646174040, 2.83319399952321921285848639021, 3.74872235057788386138427733688, 4.95416051002752486062180606059, 5.82440419129484789367904797063, 6.60029503510882269989421706911, 7.61378387688634443947355058820, 9.083934524407915191860508689998, 9.377401962677607160139664089919, 9.873543734918512250507618981673

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