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Properties

Label 2-950-25.16-c1-0-36
Degree $2$
Conductor $950$
Sign $0.633 + 0.773i$
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 + (0.154 + 0.474i)3-s + (0.309 + 0.951i)4-s + (−1.19 − 1.89i)5-s + (−0.154 + 0.474i)6-s − 2.61·7-s + (−0.309 + 0.951i)8-s + (2.22 − 1.61i)9-s + (0.145 − 2.23i)10-s + (−1.39 − 1.01i)11-s + (−0.403 + 0.293i)12-s + (1.71 − 1.24i)13-s + (−2.11 − 1.53i)14-s + (0.713 − 0.858i)15-s + (−0.809 + 0.587i)16-s + (1.79 − 5.53i)17-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (0.0890 + 0.274i)3-s + (0.154 + 0.475i)4-s + (−0.533 − 0.845i)5-s + (−0.0629 + 0.193i)6-s − 0.988·7-s + (−0.109 + 0.336i)8-s + (0.741 − 0.538i)9-s + (0.0461 − 0.705i)10-s + (−0.419 − 0.305i)11-s + (−0.116 + 0.0847i)12-s + (0.476 − 0.346i)13-s + (−0.565 − 0.410i)14-s + (0.184 − 0.221i)15-s + (−0.202 + 0.146i)16-s + (0.436 − 1.34i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.633 + 0.773i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.633 + 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.633 + 0.773i$
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: \((2,\ 950,\ (\ :1/2),\ 0.633 + 0.773i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.45784 - 0.690401i\)
\(L(\frac12)\) \(\approx\) \(1.45784 - 0.690401i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.19 + 1.89i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
good3 \( 1 + (-0.154 - 0.474i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 + 2.61T + 7T^{2} \)
11 \( 1 + (1.39 + 1.01i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-1.71 + 1.24i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.79 + 5.53i)T + (-13.7 - 9.99i)T^{2} \)
23 \( 1 + (1.05 + 0.763i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (2.90 + 8.93i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-1.91 + 5.90i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-8.17 + 5.94i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (1.22 - 0.888i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 5.61T + 43T^{2} \)
47 \( 1 + (-1.36 - 4.19i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-0.544 - 1.67i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (8.10 - 5.88i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (4.45 + 3.23i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (1.28 - 3.93i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (4.37 + 13.4i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-4.10 - 2.98i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-2.92 - 9.01i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (1.73 - 5.34i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (1.00 + 0.733i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-5.28 - 16.2i)T + (-78.4 + 57.0i)T^{2} \)
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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

−9.514639446979112764254125003178, −9.365267869223852204235094661009, −7.989333476964625965758963926707, −7.51603630636452279907487098810, −6.31463344995726486940129140357, −5.62722297889844112575657888079, −4.45667256008423521755609834151, −3.83189274974094026907876640024, −2.78119402337701459875085255493, −0.63138290457599066623029746477, 1.64342579613196331649228037910, 2.92947414494163376211032817795, 3.69783063521430481012802677573, 4.66717827586008949903390624632, 5.98057491655612509247717635339, 6.71675673521549522928864238071, 7.42558748214036117985641534187, 8.391666595983659844204683113507, 9.605479607712311295929501806474, 10.45784834754181259454143785855

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