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Properties

Label 2-960-192.131-c1-0-2
Degree $2$
Conductor $960$
Sign $-0.951 + 0.308i$
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  + (0.908 + 1.08i)2-s + (0.0753 − 1.73i)3-s + (−0.349 + 1.96i)4-s + (−0.555 + 0.831i)5-s + (1.94 − 1.49i)6-s + (−0.894 − 2.16i)7-s + (−2.45 + 1.41i)8-s + (−2.98 − 0.260i)9-s + (−1.40 + 0.153i)10-s + (−2.78 + 0.553i)11-s + (3.38 + 0.752i)12-s + (−1.64 + 1.09i)13-s + (1.52 − 2.93i)14-s + (1.39 + 1.02i)15-s + (−3.75 − 1.37i)16-s + (2.08 − 2.08i)17-s + ⋯
L(s)  = 1  + (0.642 + 0.766i)2-s + (0.0434 − 0.999i)3-s + (−0.174 + 0.984i)4-s + (−0.248 + 0.371i)5-s + (0.793 − 0.608i)6-s + (−0.338 − 0.816i)7-s + (−0.866 + 0.498i)8-s + (−0.996 − 0.0868i)9-s + (−0.444 + 0.0484i)10-s + (−0.838 + 0.166i)11-s + (0.976 + 0.217i)12-s + (−0.455 + 0.304i)13-s + (0.408 − 0.783i)14-s + (0.360 + 0.264i)15-s + (−0.939 − 0.343i)16-s + (0.506 − 0.506i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.951 + 0.308i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.951 + 0.308i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $-0.951 + 0.308i$
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: \((2,\ 960,\ (\ :1/2),\ -0.951 + 0.308i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0183481 - 0.115957i\)
\(L(\frac12)\) \(\approx\) \(0.0183481 - 0.115957i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.908 - 1.08i)T \)
3 \( 1 + (-0.0753 + 1.73i)T \)
5 \( 1 + (0.555 - 0.831i)T \)
good7 \( 1 + (0.894 + 2.16i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (2.78 - 0.553i)T + (10.1 - 4.20i)T^{2} \)
13 \( 1 + (1.64 - 1.09i)T + (4.97 - 12.0i)T^{2} \)
17 \( 1 + (-2.08 + 2.08i)T - 17iT^{2} \)
19 \( 1 + (5.32 - 3.55i)T + (7.27 - 17.5i)T^{2} \)
23 \( 1 + (2.92 - 7.05i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (-1.77 + 8.94i)T + (-26.7 - 11.0i)T^{2} \)
31 \( 1 + 5.53T + 31T^{2} \)
37 \( 1 + (-1.90 + 2.85i)T + (-14.1 - 34.1i)T^{2} \)
41 \( 1 + (0.879 + 0.364i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (9.53 - 1.89i)T + (39.7 - 16.4i)T^{2} \)
47 \( 1 + (-4.63 - 4.63i)T + 47iT^{2} \)
53 \( 1 + (-1.07 - 5.38i)T + (-48.9 + 20.2i)T^{2} \)
59 \( 1 + (-7.94 - 5.30i)T + (22.5 + 54.5i)T^{2} \)
61 \( 1 + (-0.364 + 1.83i)T + (-56.3 - 23.3i)T^{2} \)
67 \( 1 + (8.67 + 1.72i)T + (61.8 + 25.6i)T^{2} \)
71 \( 1 + (-8.56 + 3.54i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (9.84 + 4.07i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (5.52 + 5.52i)T + 79iT^{2} \)
83 \( 1 + (3.20 + 4.79i)T + (-31.7 + 76.6i)T^{2} \)
89 \( 1 + (-3.13 + 1.29i)T + (62.9 - 62.9i)T^{2} \)
97 \( 1 - 5.83iT - 97T^{2} \)
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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

−10.54291249422622828387824847372, −9.556617997925412305827735965955, −8.344100337534355748379105216246, −7.56734984719305433593836253117, −7.29509353612080416893452856180, −6.25914295854583279002381028563, −5.57687030040322084410390784812, −4.29194467360444747960110177507, −3.33323869239472079471644447572, −2.19121272973260293682819871730, 0.03961809478662713816100829347, 2.30362921601071773495440167926, 3.12628018136280998146469279694, 4.18095409333634365466901865031, 5.07419375676432510891874063022, 5.62041222065334329613143717566, 6.70814594208515354136314111424, 8.456234182916150873511885366959, 8.747772703724889789447165970839, 9.864562116504595361008370891075

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