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Properties

Label 2-950-25.11-c1-0-13
Degree $2$
Conductor $950$
Sign $0.992 + 0.118i$
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 + (0.558 − 1.71i)3-s + (0.309 − 0.951i)4-s + (−1.95 + 1.08i)5-s + (0.558 + 1.71i)6-s + 1.04·7-s + (0.309 + 0.951i)8-s + (−0.216 − 0.157i)9-s + (0.944 − 2.02i)10-s + (0.937 − 0.681i)11-s + (−1.46 − 1.06i)12-s + (−0.281 − 0.204i)13-s + (−0.847 + 0.615i)14-s + (0.771 + 3.96i)15-s + (−0.809 − 0.587i)16-s + (1.31 + 4.06i)17-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (0.322 − 0.992i)3-s + (0.154 − 0.475i)4-s + (−0.874 + 0.484i)5-s + (0.228 + 0.701i)6-s + 0.396·7-s + (0.109 + 0.336i)8-s + (−0.0721 − 0.0523i)9-s + (0.298 − 0.640i)10-s + (0.282 − 0.205i)11-s + (−0.422 − 0.306i)12-s + (−0.0779 − 0.0566i)13-s + (−0.226 + 0.164i)14-s + (0.199 + 1.02i)15-s + (−0.202 − 0.146i)16-s + (0.320 + 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.118i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 + 0.118i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.992 + 0.118i$
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: \((2,\ 950,\ (\ :1/2),\ 0.992 + 0.118i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24313 - 0.0736938i\)
\(L(\frac12)\) \(\approx\) \(1.24313 - 0.0736938i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.95 - 1.08i)T \)
19 \( 1 + (-0.309 - 0.951i)T \)
good3 \( 1 + (-0.558 + 1.71i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 - 1.04T + 7T^{2} \)
11 \( 1 + (-0.937 + 0.681i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (0.281 + 0.204i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-1.31 - 4.06i)T + (-13.7 + 9.99i)T^{2} \)
23 \( 1 + (-3.30 + 2.39i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (2.26 - 6.98i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (1.75 + 5.39i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-5.49 - 3.99i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-9.87 - 7.17i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 0.452T + 43T^{2} \)
47 \( 1 + (-2.71 + 8.34i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-2.06 + 6.34i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-6.93 - 5.03i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-5.51 + 4.00i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (2.19 + 6.75i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (-0.441 + 1.35i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-10.2 + 7.46i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-3.24 + 9.97i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-0.595 - 1.83i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (2.16 - 1.57i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (4.85 - 14.9i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.00539953634190615983033262249, −8.853443218128605269661347489033, −8.092406707755815287627267044977, −7.65225111606516967741214109827, −6.85259755526295126871773493915, −6.13161146466479230848503156694, −4.79243812458884347873226259870, −3.57416147670943575426313997680, −2.26179306008629462694363409381, −1.00332670519052782031572121362, 0.972151148506273456466014682941, 2.71039153846954217715612750844, 3.82983153554725217213662644835, 4.43552712079582500141281707071, 5.41500325132837627103991367536, 7.06661717441299544201859196042, 7.68373608765704649469453812518, 8.659348056325538844447667099598, 9.331524930110102738947659016458, 9.795453064239696178642698385770

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