Properties

Label 2-950-25.11-c1-0-4
Degree $2$
Conductor $950$
Sign $-0.0755 - 0.997i$
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 + (0.272 − 0.839i)3-s + (0.309 − 0.951i)4-s + (0.404 − 2.19i)5-s + (0.272 + 0.839i)6-s − 4.83·7-s + (0.309 + 0.951i)8-s + (1.79 + 1.30i)9-s + (0.964 + 2.01i)10-s + (−3.36 + 2.44i)11-s + (−0.713 − 0.518i)12-s + (1.69 + 1.22i)13-s + (3.91 − 2.84i)14-s + (−1.73 − 0.939i)15-s + (−0.809 − 0.587i)16-s + (0.451 + 1.38i)17-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (0.157 − 0.484i)3-s + (0.154 − 0.475i)4-s + (0.181 − 0.983i)5-s + (0.111 + 0.342i)6-s − 1.82·7-s + (0.109 + 0.336i)8-s + (0.599 + 0.435i)9-s + (0.305 + 0.637i)10-s + (−1.01 + 0.736i)11-s + (−0.206 − 0.149i)12-s + (0.469 + 0.340i)13-s + (1.04 − 0.759i)14-s + (−0.448 − 0.242i)15-s + (−0.202 − 0.146i)16-s + (0.109 + 0.336i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0755 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0755 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.0755 - 0.997i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (761, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.0755 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.402166 + 0.433781i\)
\(L(\frac12)\) \(\approx\) \(0.402166 + 0.433781i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.404 + 2.19i)T \)
19 \( 1 + (-0.309 - 0.951i)T \)
good3 \( 1 + (-0.272 + 0.839i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 + 4.83T + 7T^{2} \)
11 \( 1 + (3.36 - 2.44i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-1.69 - 1.22i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-0.451 - 1.38i)T + (-13.7 + 9.99i)T^{2} \)
23 \( 1 + (3.34 - 2.42i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (-0.683 + 2.10i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-2.45 - 7.55i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-2.22 - 1.61i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-8.48 - 6.16i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 9.95T + 43T^{2} \)
47 \( 1 + (-0.642 + 1.97i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (1.84 - 5.67i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-4.19 - 3.04i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (0.825 - 0.599i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (0.0474 + 0.146i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (-2.75 + 8.47i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-2.07 + 1.50i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (2.24 - 6.92i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-4.66 - 14.3i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (15.0 - 10.9i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (0.505 - 1.55i)T + (-78.4 - 57.0i)T^{2} \)
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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

−9.770077080733453326340964320718, −9.695158106993169504179046005078, −8.473382093175699063342780393795, −7.83069552716025368173078881889, −6.89074272652876405167379174455, −6.19855566585124648649029519021, −5.21730753274656460453226105415, −4.07933765874300271180543573516, −2.60711826879666891121267906541, −1.33328426805998101889006341001, 0.33778622912697416519713042334, 2.57534591321920123724517307900, 3.22893451910605520729191826488, 3.98428462812520876993699781673, 5.74545530594340602221789656224, 6.48926788424572119143926954643, 7.25730067638650858242402573628, 8.289592244633573397051495246987, 9.369004264311555813992550352736, 9.878431009216855777032621135717

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