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Properties

Label 2-960-12.11-c3-0-85
Degree $2$
Conductor $960$
Sign $-0.474 + 0.880i$
Analytic cond. $56.6418$
Root an. cond. $7.52607$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.57 + 2.46i)3-s + 5i·5-s − 35.9i·7-s + (14.8 + 22.5i)9-s − 56.8·11-s + 48.7·13-s + (−12.3 + 22.8i)15-s − 18.2i·17-s + 20.4i·19-s + (88.5 − 164. i)21-s − 161.·23-s − 25·25-s + (12.4 + 139. i)27-s − 68.7i·29-s − 166. i·31-s + ⋯
L(s)  = 1  + (0.880 + 0.474i)3-s + 0.447i·5-s − 1.94i·7-s + (0.550 + 0.835i)9-s − 1.55·11-s + 1.03·13-s + (−0.212 + 0.393i)15-s − 0.260i·17-s + 0.246i·19-s + (0.920 − 1.70i)21-s − 1.46·23-s − 0.200·25-s + (0.0884 + 0.996i)27-s − 0.440i·29-s − 0.964i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.474 + 0.880i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.474 + 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $-0.474 + 0.880i$
Analytic conductor: \(56.6418\)
Root analytic conductor: \(7.52607\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :3/2),\ -0.474 + 0.880i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.320540148\)
\(L(\frac12)\) \(\approx\) \(1.320540148\)
\(L(\frac{5}{2})\) 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 \( 1 + (-4.57 - 2.46i)T \)
5 \( 1 - 5iT \)
good7 \( 1 + 35.9iT - 343T^{2} \)
11 \( 1 + 56.8T + 1.33e3T^{2} \)
13 \( 1 - 48.7T + 2.19e3T^{2} \)
17 \( 1 + 18.2iT - 4.91e3T^{2} \)
19 \( 1 - 20.4iT - 6.85e3T^{2} \)
23 \( 1 + 161.T + 1.21e4T^{2} \)
29 \( 1 + 68.7iT - 2.43e4T^{2} \)
31 \( 1 + 166. iT - 2.97e4T^{2} \)
37 \( 1 - 135.T + 5.06e4T^{2} \)
41 \( 1 - 32.0iT - 6.89e4T^{2} \)
43 \( 1 + 259. iT - 7.95e4T^{2} \)
47 \( 1 + 396.T + 1.03e5T^{2} \)
53 \( 1 + 477. iT - 1.48e5T^{2} \)
59 \( 1 + 268.T + 2.05e5T^{2} \)
61 \( 1 + 501.T + 2.26e5T^{2} \)
67 \( 1 - 327. iT - 3.00e5T^{2} \)
71 \( 1 + 1.06e3T + 3.57e5T^{2} \)
73 \( 1 - 945.T + 3.89e5T^{2} \)
79 \( 1 + 376. iT - 4.93e5T^{2} \)
83 \( 1 + 506.T + 5.71e5T^{2} \)
89 \( 1 + 944. iT - 7.04e5T^{2} \)
97 \( 1 - 643.T + 9.12e5T^{2} \)
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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

−9.651108951941061479035461114710, −8.215475812467555887982317951081, −7.85580412184939062076842565422, −7.11002761031463967531650436559, −5.94923138859099772157936620733, −4.62450424185070434498214008025, −3.89599026442586113282735562258, −3.10607859420283341632086446680, −1.83860982707800935233942819015, −0.26877820870502509010531260975, 1.56429720055308410471991275545, 2.48836906370611224725758179071, 3.27045811459365971069996749654, 4.71811930227172975890172988992, 5.72784543187580635169252133249, 6.34296731567238309813759133087, 7.81851737108660376857913217323, 8.264820646609267897079655076135, 8.911463508991813454093952494774, 9.619344176820055400433897613129

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