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Properties

Label 2-950-95.27-c1-0-10
Degree $2$
Conductor $950$
Sign $0.349 - 0.936i$
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  + (0.258 + 0.965i)2-s + (−1.47 + 0.394i)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 + (0.951 − 3.55i)13-s + (1.19 − 2.07i)14-s + (0.500 − 0.866i)16-s + (−0.153 + 0.0410i)17-s + (−0.479 − 0.479i)18-s + (−0.533 + 4.32i)19-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (−0.849 + 0.227i)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 + (0.263 − 0.985i)13-s + (0.320 − 0.555i)14-s + (0.125 − 0.216i)16-s + (−0.0371 + 0.00995i)17-s + (−0.112 − 0.112i)18-s + (−0.122 + 0.992i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.349 - 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.349 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.349 - 0.936i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (407, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.349 - 0.936i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.924194 + 0.641539i\)
\(L(\frac12)\) \(\approx\) \(0.924194 + 0.641539i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 - 0.965i)T \)
5 \( 1 \)
19 \( 1 + (0.533 - 4.32i)T \)
good3 \( 1 + (1.47 - 0.394i)T + (2.59 - 1.5i)T^{2} \)
7 \( 1 + (1.69 + 1.69i)T + 7iT^{2} \)
11 \( 1 - 4.92T + 11T^{2} \)
13 \( 1 + (-0.951 + 3.55i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (0.153 - 0.0410i)T + (14.7 - 8.5i)T^{2} \)
23 \( 1 + (-2.31 - 0.620i)T + (19.9 + 11.5i)T^{2} \)
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 \( 1 + (-1.64 - 6.15i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (0.702 - 2.62i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (1.61 - 6.03i)T + (-45.8 - 26.5i)T^{2} \)
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 \( 1 + (-8.84 - 2.36i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-3.64 - 2.10i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.847 + 3.16i)T + (-63.2 + 36.5i)T^{2} \)
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 \( 1 + (-4.52 - 16.8i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.22609522796805504524713401836, −9.386437540485174695871903518372, −8.450714275095478934421522524261, −7.53124366469114380851517227629, −6.48276150211138985226106061621, −6.09915296426625757308115207156, −5.09353174019013016613267285090, −4.10341809100476755608529879930, −3.19472267882153391268351265288, −0.947530600849753441023504438826, 0.792281381743498270609466903639, 2.26377406903503483784042583077, 3.50523332879263580197194363886, 4.49985561181589851288958398768, 5.56806178422384886962218031799, 6.47166563789807214034828670462, 6.85623969120592269785874923303, 8.666493819924181115825861251872, 9.089810258787461820337760368747, 9.876630896875070759961731485883

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