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Properties

Label 2-950-95.12-c1-0-13
Degree $2$
Conductor $950$
Sign $0.302 - 0.953i$
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  + (0.965 + 0.258i)2-s + (−0.394 + 1.47i)3-s + (0.866 + 0.499i)4-s + (−0.761 + 1.31i)6-s + (−1.69 − 1.69i)7-s + (0.707 + 0.707i)8-s + (0.586 + 0.338i)9-s + 4.92·11-s + (−1.07 + 1.07i)12-s + (3.55 − 0.951i)13-s + (−1.19 − 2.07i)14-s + (0.500 + 0.866i)16-s + (0.0410 − 0.153i)17-s + (0.479 + 0.479i)18-s + (0.533 + 4.32i)19-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (−0.227 + 0.849i)3-s + (0.433 + 0.249i)4-s + (−0.311 + 0.538i)6-s + (−0.640 − 0.640i)7-s + (0.249 + 0.249i)8-s + (0.195 + 0.112i)9-s + 1.48·11-s + (−0.311 + 0.311i)12-s + (0.985 − 0.263i)13-s + (−0.320 − 0.555i)14-s + (0.125 + 0.216i)16-s + (0.00995 − 0.0371i)17-s + (0.112 + 0.112i)18-s + (0.122 + 0.992i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.302 - 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.302 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.302 - 0.953i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.302 - 0.953i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.92184 + 1.40565i\)
\(L(\frac12)\) \(\approx\) \(1.92184 + 1.40565i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 - 0.258i)T \)
5 \( 1 \)
19 \( 1 + (-0.533 - 4.32i)T \)
good3 \( 1 + (0.394 - 1.47i)T + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + (1.69 + 1.69i)T + 7iT^{2} \)
11 \( 1 - 4.92T + 11T^{2} \)
13 \( 1 + (-3.55 + 0.951i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (-0.0410 + 0.153i)T + (-14.7 - 8.5i)T^{2} \)
23 \( 1 + (0.620 + 2.31i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (4.62 - 8.01i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.10iT - 31T^{2} \)
37 \( 1 + (-2.44 + 2.44i)T - 37iT^{2} \)
41 \( 1 + (-6.51 + 3.75i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.15 + 1.64i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (-2.62 + 0.702i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (6.03 - 1.61i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (1.77 + 3.07i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.36 + 4.09i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.36 - 8.84i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-3.64 + 2.10i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-3.16 - 0.847i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (7.39 + 12.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.27 + 2.27i)T - 83iT^{2} \)
89 \( 1 + (-3.93 + 6.80i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-16.8 - 4.52i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.39437511085978399214737747238, −9.460404962027047916372821185985, −8.679680890270953109621475911229, −7.43271403673658006518327164924, −6.64152444198354595875537037246, −5.86015677643511373970931721348, −4.82374200492189388872449595120, −3.73956384999081504682364244409, −3.57660995854250565928146388284, −1.49806448036998910898296822744, 1.08889143089443960000736649416, 2.26175392625906980573004428689, 3.58315907109174326696702710848, 4.38086471262631724772112628904, 5.93035379866314920690889694705, 6.25610083623753698079471922851, 7.01465028736596911140080040952, 8.040588542382906860336008268056, 9.323142034547831427424770501171, 9.611709823645615439884369015186

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