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Properties

Label 2-960-15.14-c2-0-19
Degree $2$
Conductor $960$
Sign $-0.999 - 0.0169i$
Analytic cond. $26.1581$
Root an. cond. $5.11449$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.49 + 1.67i)3-s + (−4.19 + 2.71i)5-s + 12.7i·7-s + (3.41 + 8.32i)9-s + 12.6i·11-s − 7.44i·13-s + (−14.9 − 0.254i)15-s + 14.0·17-s − 31.0·19-s + (−21.3 + 31.8i)21-s + 7.50·23-s + (10.2 − 22.7i)25-s + (−5.40 + 26.4i)27-s − 15.7i·29-s + 20.4·31-s + ⋯
L(s)  = 1  + (0.830 + 0.557i)3-s + (−0.839 + 0.542i)5-s + 1.82i·7-s + (0.379 + 0.925i)9-s + 1.14i·11-s − 0.572i·13-s + (−0.999 − 0.0169i)15-s + 0.826·17-s − 1.63·19-s + (−1.01 + 1.51i)21-s + 0.326·23-s + (0.410 − 0.911i)25-s + (−0.200 + 0.979i)27-s − 0.542i·29-s + 0.660·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0169i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.999 - 0.0169i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $-0.999 - 0.0169i$
Analytic conductor: \(26.1581\)
Root analytic conductor: \(5.11449\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1),\ -0.999 - 0.0169i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.656313071\)
\(L(\frac12)\) \(\approx\) \(1.656313071\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-2.49 - 1.67i)T \)
5 \( 1 + (4.19 - 2.71i)T \)
good7 \( 1 - 12.7iT - 49T^{2} \)
11 \( 1 - 12.6iT - 121T^{2} \)
13 \( 1 + 7.44iT - 169T^{2} \)
17 \( 1 - 14.0T + 289T^{2} \)
19 \( 1 + 31.0T + 361T^{2} \)
23 \( 1 - 7.50T + 529T^{2} \)
29 \( 1 + 15.7iT - 841T^{2} \)
31 \( 1 - 20.4T + 961T^{2} \)
37 \( 1 + 12.9iT - 1.36e3T^{2} \)
41 \( 1 + 13.8iT - 1.68e3T^{2} \)
43 \( 1 + 30.0iT - 1.84e3T^{2} \)
47 \( 1 - 20.2T + 2.20e3T^{2} \)
53 \( 1 - 29.1T + 2.80e3T^{2} \)
59 \( 1 - 47.6iT - 3.48e3T^{2} \)
61 \( 1 + 43.0T + 3.72e3T^{2} \)
67 \( 1 + 0.630iT - 4.48e3T^{2} \)
71 \( 1 - 90.4iT - 5.04e3T^{2} \)
73 \( 1 + 46.2iT - 5.32e3T^{2} \)
79 \( 1 + 37.9T + 6.24e3T^{2} \)
83 \( 1 - 80.2T + 6.88e3T^{2} \)
89 \( 1 - 140. iT - 7.92e3T^{2} \)
97 \( 1 - 10.3iT - 9.40e3T^{2} \)
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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

−10.18946115189997875672249345412, −9.323278752617751913889112735259, −8.532777830497762725151454085525, −7.988446971406172732637675207195, −7.03206398318403209089043948813, −5.86786670817692764877933762907, −4.85488637740253127945972868781, −3.92882436515835184878200027518, −2.79382836578902993457800539837, −2.17862442044893926451601299720, 0.48848405543140484502904415663, 1.38832570347138440310073086419, 3.16248823832609942538381991201, 3.90357258414986273644369417443, 4.63049017081963331076685328046, 6.29341227624917689342557838772, 7.04911069614556366425228668846, 7.85791458634212685290912150424, 8.371794499422086653566600714832, 9.206698987890769844128107220887

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