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Properties

Label 2-950-95.88-c1-0-14
Degree $2$
Conductor $950$
Sign $-0.0398 - 0.999i$
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 + (2.51 + 0.672i)3-s + (−0.866 − 0.499i)4-s + (−1.29 + 2.25i)6-s + (−0.692 + 0.692i)7-s + (0.707 − 0.707i)8-s + (3.25 + 1.87i)9-s + 4.06·11-s + (−1.83 − 1.83i)12-s + (1.66 + 6.19i)13-s + (−0.489 − 0.847i)14-s + (0.500 + 0.866i)16-s + (−2.50 − 0.670i)17-s + (−2.65 + 2.65i)18-s + (0.843 − 4.27i)19-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (1.44 + 0.388i)3-s + (−0.433 − 0.249i)4-s + (−0.530 + 0.918i)6-s + (−0.261 + 0.261i)7-s + (0.249 − 0.249i)8-s + (1.08 + 0.625i)9-s + 1.22·11-s + (−0.530 − 0.530i)12-s + (0.460 + 1.71i)13-s + (−0.130 − 0.226i)14-s + (0.125 + 0.216i)16-s + (−0.606 − 0.162i)17-s + (−0.625 + 0.625i)18-s + (0.193 − 0.981i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0398 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0398 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.0398 - 0.999i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (943, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.0398 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.61401 + 1.67974i\)
\(L(\frac12)\) \(\approx\) \(1.61401 + 1.67974i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.843 + 4.27i)T \)
good3 \( 1 + (-2.51 - 0.672i)T + (2.59 + 1.5i)T^{2} \)
7 \( 1 + (0.692 - 0.692i)T - 7iT^{2} \)
11 \( 1 - 4.06T + 11T^{2} \)
13 \( 1 + (-1.66 - 6.19i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (2.50 + 0.670i)T + (14.7 + 8.5i)T^{2} \)
23 \( 1 + (-3.52 + 0.943i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (1.89 - 3.28i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 5.30iT - 31T^{2} \)
37 \( 1 + (-2.58 - 2.58i)T + 37iT^{2} \)
41 \( 1 + (7.09 - 4.09i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.23 - 4.61i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (-1.68 - 6.29i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (0.565 + 2.10i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (2.99 + 5.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.43 - 7.68i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-9.25 + 2.47i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-6.72 + 3.88i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-4.02 + 15.0i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (3.49 + 6.05i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.44 - 4.44i)T + 83iT^{2} \)
89 \( 1 + (-1.61 + 2.80i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.62 - 9.80i)T + (-84.0 - 48.5i)T^{2} \)
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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

−9.551940828423274106143796296485, −9.251139963569640848015893930430, −8.846888466121467514933065599915, −7.88632219748923667920733455010, −6.82832408233486029774412178133, −6.36176649145341486387920207073, −4.71842767958999882755473872173, −4.06224544249351877615169134737, −2.99350940194915525428490148109, −1.68559887319350803560865917010, 1.12195960009795238921813127840, 2.27973556417531075061569666217, 3.47769336574046991817949965383, 3.75739275268919080833030392959, 5.37971251547618863392684726294, 6.66962166360148775306039560185, 7.57703987477545130242699486839, 8.411154690349787289092611344994, 8.868890725455654735708304658329, 9.778733031831504551984399654919

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