Properties

Label 2-960-192.131-c1-0-114
Degree $2$
Conductor $960$
Sign $0.273 + 0.961i$
Analytic cond. $7.66563$
Root an. cond. $2.76868$
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  + (1.30 − 0.536i)2-s + (0.342 + 1.69i)3-s + (1.42 − 1.40i)4-s + (0.555 − 0.831i)5-s + (1.35 + 2.03i)6-s + (−1.78 − 4.30i)7-s + (1.10 − 2.60i)8-s + (−2.76 + 1.16i)9-s + (0.280 − 1.38i)10-s + (−5.16 + 1.02i)11-s + (2.87 + 1.93i)12-s + (4.07 − 2.72i)13-s + (−4.64 − 4.67i)14-s + (1.60 + 0.658i)15-s + (0.0528 − 3.99i)16-s + (4.90 − 4.90i)17-s + ⋯
L(s)  = 1  + (0.925 − 0.379i)2-s + (0.197 + 0.980i)3-s + (0.711 − 0.702i)4-s + (0.248 − 0.371i)5-s + (0.554 + 0.831i)6-s + (−0.673 − 1.62i)7-s + (0.391 − 0.920i)8-s + (−0.921 + 0.387i)9-s + (0.0886 − 0.438i)10-s + (−1.55 + 0.309i)11-s + (0.829 + 0.558i)12-s + (1.13 − 0.755i)13-s + (−1.24 − 1.24i)14-s + (0.413 + 0.170i)15-s + (0.0132 − 0.999i)16-s + (1.18 − 1.18i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.273 + 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.273 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $0.273 + 0.961i$
Analytic conductor: \(7.66563\)
Root analytic conductor: \(2.76868\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1/2),\ 0.273 + 0.961i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.10875 - 1.59244i\)
\(L(\frac12)\) \(\approx\) \(2.10875 - 1.59244i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.30 + 0.536i)T \)
3 \( 1 + (-0.342 - 1.69i)T \)
5 \( 1 + (-0.555 + 0.831i)T \)
good7 \( 1 + (1.78 + 4.30i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (5.16 - 1.02i)T + (10.1 - 4.20i)T^{2} \)
13 \( 1 + (-4.07 + 2.72i)T + (4.97 - 12.0i)T^{2} \)
17 \( 1 + (-4.90 + 4.90i)T - 17iT^{2} \)
19 \( 1 + (0.102 - 0.0687i)T + (7.27 - 17.5i)T^{2} \)
23 \( 1 + (-0.763 + 1.84i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (1.79 - 9.04i)T + (-26.7 - 11.0i)T^{2} \)
31 \( 1 - 5.71T + 31T^{2} \)
37 \( 1 + (3.02 - 4.52i)T + (-14.1 - 34.1i)T^{2} \)
41 \( 1 + (-8.86 - 3.67i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (3.35 - 0.667i)T + (39.7 - 16.4i)T^{2} \)
47 \( 1 + (-4.28 - 4.28i)T + 47iT^{2} \)
53 \( 1 + (-0.984 - 4.95i)T + (-48.9 + 20.2i)T^{2} \)
59 \( 1 + (7.95 + 5.31i)T + (22.5 + 54.5i)T^{2} \)
61 \( 1 + (-1.08 + 5.45i)T + (-56.3 - 23.3i)T^{2} \)
67 \( 1 + (-0.578 - 0.115i)T + (61.8 + 25.6i)T^{2} \)
71 \( 1 + (-2.90 + 1.20i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (-2.91 - 1.20i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (0.899 + 0.899i)T + 79iT^{2} \)
83 \( 1 + (-2.63 - 3.93i)T + (-31.7 + 76.6i)T^{2} \)
89 \( 1 + (-2.52 + 1.04i)T + (62.9 - 62.9i)T^{2} \)
97 \( 1 + 4.72iT - 97T^{2} \)
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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

−10.15048432731232940592149101905, −9.464278834752087643141436545127, −8.070697482818577386241786990395, −7.30081345511809848331013616952, −6.10267891515583555015730962521, −5.16848695552168215105066202004, −4.57197285565141844024065091035, −3.40768999684144063085774765023, −2.93704532248221736975910422948, −0.885075843689633774213016037677, 2.05166778176457242643585778991, 2.76489936565196507986079431776, 3.68602682908540033373628105348, 5.52351829762753041560267845924, 5.86114745714168195651818983489, 6.46475036236371299504146303303, 7.66046475158335851128579444720, 8.270083270566177518337013134183, 9.048414085463679712194232468287, 10.35155221744700384751930222178

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