Properties

Label 2-950-19.7-c1-0-3
Degree $2$
Conductor $950$
Sign $0.932 + 0.360i$
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  + (−0.5 − 0.866i)2-s + (−1.58 − 2.75i)3-s + (−0.499 + 0.866i)4-s + (−1.58 + 2.75i)6-s − 4.17·7-s + 0.999·8-s + (−3.54 + 6.14i)9-s − 3.90·11-s + 3.17·12-s + (−0.239 + 0.414i)13-s + (2.08 + 3.61i)14-s + (−0.5 − 0.866i)16-s + (0.245 + 0.424i)17-s + 7.09·18-s + (2.74 − 3.38i)19-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.916 − 1.58i)3-s + (−0.249 + 0.433i)4-s + (−0.648 + 1.12i)6-s − 1.57·7-s + 0.353·8-s + (−1.18 + 2.04i)9-s − 1.17·11-s + 0.916·12-s + (−0.0664 + 0.115i)13-s + (0.558 + 0.966i)14-s + (−0.125 − 0.216i)16-s + (0.0594 + 0.102i)17-s + 1.67·18-s + (0.628 − 0.777i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.932 + 0.360i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.932 + 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.932 + 0.360i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (501, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.932 + 0.360i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.270962 - 0.0506002i\)
\(L(\frac12)\) \(\approx\) \(0.270962 - 0.0506002i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 \)
19 \( 1 + (-2.74 + 3.38i)T \)
good3 \( 1 + (1.58 + 2.75i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + 4.17T + 7T^{2} \)
11 \( 1 + 3.90T + 11T^{2} \)
13 \( 1 + (0.239 - 0.414i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.245 - 0.424i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-2.84 + 4.93i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.21 - 2.10i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.42T + 31T^{2} \)
37 \( 1 + 8.47T + 37T^{2} \)
41 \( 1 + (-0.254 - 0.441i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.91 - 5.04i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.485 + 0.841i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.77 - 4.81i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.82 - 6.62i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.65 + 8.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.809 - 1.40i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.937 - 1.62i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.35 - 4.07i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.19 + 5.54i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 7.96T + 83T^{2} \)
89 \( 1 + (-2.76 + 4.78i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.24 - 12.5i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.25616478065153796372983276088, −9.189123522132042917717280330474, −8.226336269877036901088121359659, −7.26560816277112335412248125685, −6.76794770891426138608554856496, −5.86070199551607035816769249159, −4.91910170519139428814643336256, −3.12078680930285156551149169133, −2.36074436197290490234722644262, −0.814864309859528319226307051989, 0.23338922891995296613710341312, 3.09597177905790571359326024090, 3.88043378282599993466355818549, 5.18501216753764824530448976397, 5.59215199506106983485731195687, 6.46856161780979382460769201904, 7.44972347661551928760344577630, 8.703623429545677905720031819504, 9.547599998173975386232420225852, 10.01333962600850893000700863345

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