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Properties

Label 2-950-95.4-c1-0-13
Degree $2$
Conductor $950$
Sign $0.724 - 0.689i$
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  + (−0.342 + 0.939i)2-s + (−1.28 + 0.227i)3-s + (−0.766 − 0.642i)4-s + (0.227 − 1.28i)6-s + (1.93 + 1.11i)7-s + (0.866 − 0.500i)8-s + (−1.20 + 0.440i)9-s + (−2.90 − 5.03i)11-s + (1.13 + 0.654i)12-s + (2.79 + 0.492i)13-s + (−1.70 + 1.43i)14-s + (0.173 + 0.984i)16-s + (−0.366 + 1.00i)17-s − 1.28i·18-s + (2.13 + 3.80i)19-s + ⋯
L(s)  = 1  + (−0.241 + 0.664i)2-s + (−0.744 + 0.131i)3-s + (−0.383 − 0.321i)4-s + (0.0927 − 0.526i)6-s + (0.730 + 0.421i)7-s + (0.306 − 0.176i)8-s + (−0.403 + 0.146i)9-s + (−0.875 − 1.51i)11-s + (0.327 + 0.188i)12-s + (0.774 + 0.136i)13-s + (−0.456 + 0.383i)14-s + (0.0434 + 0.246i)16-s + (−0.0888 + 0.244i)17-s − 0.303i·18-s + (0.488 + 0.872i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.724 - 0.689i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.724 - 0.689i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.724 - 0.689i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.724 - 0.689i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.935823 + 0.374298i\)
\(L(\frac12)\) \(\approx\) \(0.935823 + 0.374298i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.342 - 0.939i)T \)
5 \( 1 \)
19 \( 1 + (-2.13 - 3.80i)T \)
good3 \( 1 + (1.28 - 0.227i)T + (2.81 - 1.02i)T^{2} \)
7 \( 1 + (-1.93 - 1.11i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.90 + 5.03i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.79 - 0.492i)T + (12.2 + 4.44i)T^{2} \)
17 \( 1 + (0.366 - 1.00i)T + (-13.0 - 10.9i)T^{2} \)
23 \( 1 + (-2.90 + 3.46i)T + (-3.99 - 22.6i)T^{2} \)
29 \( 1 + (0.483 - 0.175i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-3.47 + 6.01i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 2.58iT - 37T^{2} \)
41 \( 1 + (0.665 + 3.77i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (-5.38 - 6.41i)T + (-7.46 + 42.3i)T^{2} \)
47 \( 1 + (-3.77 - 10.3i)T + (-36.0 + 30.2i)T^{2} \)
53 \( 1 + (1.89 - 2.26i)T + (-9.20 - 52.1i)T^{2} \)
59 \( 1 + (-6.57 - 2.39i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-11.2 - 9.47i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (-2.77 - 7.63i)T + (-51.3 + 43.0i)T^{2} \)
71 \( 1 + (-5.09 + 4.27i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (-3.34 + 0.589i)T + (68.5 - 24.9i)T^{2} \)
79 \( 1 + (-0.901 - 5.11i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-14.3 - 8.27i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.355 - 2.01i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (1.70 - 4.68i)T + (-74.3 - 62.3i)T^{2} \)
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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

−10.28437339654924740593409062361, −9.062160066325210461002013759933, −8.291330032391273730241287295052, −7.902778647386787496368158732576, −6.48188281687447741517481195833, −5.70719386660240825238198905113, −5.36295591015818248591401044606, −4.12220048676098649919817537579, −2.68795148683212781040890477331, −0.845465347689904375088896979959, 0.888048849893681322957158289708, 2.22656941721888111989503385769, 3.49594515402784992672546687289, 4.88638320285252705911992588896, 5.18873431663279769097115572380, 6.65743469458130847601452235484, 7.42777377803074978975989964155, 8.323341170661981051687764983417, 9.236959006661015186370601236847, 10.17159235058487452602145904117

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