Properties

Label 2-960-120.29-c2-0-39
Degree $2$
Conductor $960$
Sign $0.482 + 0.875i$
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  + (−1.31 − 2.69i)3-s + (−1.53 − 4.75i)5-s + 12.2i·7-s + (−5.56 + 7.07i)9-s − 3.02·11-s − 13.7·13-s + (−10.8 + 10.3i)15-s + 25.8·17-s − 14.0i·19-s + (33.0 − 16.0i)21-s + 17.8·23-s + (−20.2 + 14.6i)25-s + (26.3 + 5.72i)27-s − 27.5·29-s + 33.8·31-s + ⋯
L(s)  = 1  + (−0.437 − 0.899i)3-s + (−0.307 − 0.951i)5-s + 1.74i·7-s + (−0.617 + 0.786i)9-s − 0.274·11-s − 1.05·13-s + (−0.721 + 0.692i)15-s + 1.52·17-s − 0.741i·19-s + (1.57 − 0.764i)21-s + 0.775·23-s + (−0.810 + 0.585i)25-s + (0.977 + 0.211i)27-s − 0.951·29-s + 1.09·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.482 + 0.875i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.482 + 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $0.482 + 0.875i$
Analytic conductor: \(26.1581\)
Root analytic conductor: \(5.11449\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (929, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1),\ 0.482 + 0.875i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.257766715\)
\(L(\frac12)\) \(\approx\) \(1.257766715\)
\(L(2)\) 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 \( 1 + (1.31 + 2.69i)T \)
5 \( 1 + (1.53 + 4.75i)T \)
good7 \( 1 - 12.2iT - 49T^{2} \)
11 \( 1 + 3.02T + 121T^{2} \)
13 \( 1 + 13.7T + 169T^{2} \)
17 \( 1 - 25.8T + 289T^{2} \)
19 \( 1 + 14.0iT - 361T^{2} \)
23 \( 1 - 17.8T + 529T^{2} \)
29 \( 1 + 27.5T + 841T^{2} \)
31 \( 1 - 33.8T + 961T^{2} \)
37 \( 1 - 14.9T + 1.36e3T^{2} \)
41 \( 1 + 69.4iT - 1.68e3T^{2} \)
43 \( 1 - 65.9T + 1.84e3T^{2} \)
47 \( 1 - 11.8T + 2.20e3T^{2} \)
53 \( 1 - 63.3iT - 2.80e3T^{2} \)
59 \( 1 - 28.6T + 3.48e3T^{2} \)
61 \( 1 + 2.88iT - 3.72e3T^{2} \)
67 \( 1 + 69.0T + 4.48e3T^{2} \)
71 \( 1 + 69.9iT - 5.04e3T^{2} \)
73 \( 1 + 113. iT - 5.32e3T^{2} \)
79 \( 1 - 109.T + 6.24e3T^{2} \)
83 \( 1 + 16.4iT - 6.88e3T^{2} \)
89 \( 1 + 66.4iT - 7.92e3T^{2} \)
97 \( 1 + 62.9iT - 9.40e3T^{2} \)
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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

−9.351138732122131337896952702256, −8.910660979226371445375356048566, −7.893180297927378435633087866613, −7.37216246451475854263128890130, −5.99810072312752984219644279551, −5.43054541960546105420844292261, −4.77546009956191657664208586138, −2.94950251826700403530695075814, −2.03188669280047971274350858869, −0.62808751006529309048750157580, 0.811116977754142292493459512890, 2.91916457816755726875372466506, 3.74392305358445540692061190486, 4.51311664474429121142635585102, 5.57639747752234652084268278207, 6.64611147991487234969322215855, 7.45471599130343480772989233036, 8.033286652827765161370952086417, 9.733997755812148114159418627008, 9.959455204805752714126406706733

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