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Properties

Label 2-950-25.16-c1-0-35
Degree $2$
Conductor $950$
Sign $0.284 + 0.958i$
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  + (0.809 + 0.587i)2-s + (−0.734 − 2.26i)3-s + (0.309 + 0.951i)4-s + (−0.796 − 2.08i)5-s + (0.734 − 2.26i)6-s + 3.12·7-s + (−0.309 + 0.951i)8-s + (−2.14 + 1.55i)9-s + (0.583 − 2.15i)10-s + (3.72 + 2.70i)11-s + (1.92 − 1.39i)12-s + (2.95 − 2.14i)13-s + (2.53 + 1.83i)14-s + (−4.13 + 3.33i)15-s + (−0.809 + 0.587i)16-s + (1.30 − 4.00i)17-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (−0.424 − 1.30i)3-s + (0.154 + 0.475i)4-s + (−0.356 − 0.934i)5-s + (0.299 − 0.923i)6-s + 1.18·7-s + (−0.109 + 0.336i)8-s + (−0.715 + 0.519i)9-s + (0.184 − 0.682i)10-s + (1.12 + 0.815i)11-s + (0.555 − 0.403i)12-s + (0.820 − 0.596i)13-s + (0.676 + 0.491i)14-s + (−1.06 + 0.861i)15-s + (−0.202 + 0.146i)16-s + (0.315 − 0.972i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.284 + 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.284 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.284 + 0.958i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.284 + 0.958i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.72365 - 1.28659i\)
\(L(\frac12)\) \(\approx\) \(1.72365 - 1.28659i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 - 0.587i)T \)
5 \( 1 + (0.796 + 2.08i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
good3 \( 1 + (0.734 + 2.26i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 - 3.12T + 7T^{2} \)
11 \( 1 + (-3.72 - 2.70i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-2.95 + 2.14i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.30 + 4.00i)T + (-13.7 - 9.99i)T^{2} \)
23 \( 1 + (3.77 + 2.74i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-2.84 - 8.75i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-1.36 + 4.19i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (0.540 - 0.392i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-3.18 + 2.31i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 5.90T + 43T^{2} \)
47 \( 1 + (1.18 + 3.66i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (3.98 + 12.2i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-5.88 + 4.27i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (5.28 + 3.83i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (2.55 - 7.85i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (-0.958 - 2.94i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-6.98 - 5.07i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-4.71 - 14.5i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (0.629 - 1.93i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (10.2 + 7.44i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-4.47 - 13.7i)T + (-78.4 + 57.0i)T^{2} \)
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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

−9.764619777470303305039562741559, −8.522546616160609875936497505444, −8.112697252115942953237544911864, −7.23032625151005728948322740215, −6.54402760630189446772724081447, −5.49305047527490834215612862077, −4.78100223615993511048392452165, −3.77710259440777121319505740641, −1.93251476275263251770421893362, −1.02213885919018785777507206157, 1.62015978856919145884928750123, 3.27753257442531979051823943170, 4.05396962102021322796808995986, 4.55504495999420400526461005929, 5.88643065887303587341706969606, 6.33700318090539309472912600676, 7.77370367623119909189168295561, 8.684107660854559318386962836315, 9.670753481591846347320961509916, 10.52130339212751610655579901235

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