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Properties

Label 2-960-40.3-c1-0-9
Degree $2$
Conductor $960$
Sign $0.957 - 0.287i$
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  + (0.707 + 0.707i)3-s + (−2.12 + 0.707i)5-s + (−1.44 − 1.44i)7-s + 1.00i·9-s + 4.87·11-s + (4.87 − 4.87i)13-s + (−2 − 0.999i)15-s + (−2 + 2i)17-s + 0.635i·19-s − 2.04i·21-s + (−1.55 + 1.55i)23-s + (3.99 − 3i)25-s + (−0.707 + 0.707i)27-s + 4.24·29-s + 6.89i·31-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (−0.948 + 0.316i)5-s + (−0.547 − 0.547i)7-s + 0.333i·9-s + 1.47·11-s + (1.35 − 1.35i)13-s + (−0.516 − 0.258i)15-s + (−0.485 + 0.485i)17-s + 0.145i·19-s − 0.447i·21-s + (−0.323 + 0.323i)23-s + (0.799 − 0.600i)25-s + (−0.136 + 0.136i)27-s + 0.787·29-s + 1.23i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.287i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 - 0.287i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $0.957 - 0.287i$
Analytic conductor: \(7.66563\)
Root analytic conductor: \(2.76868\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1/2),\ 0.957 - 0.287i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.61318 + 0.237028i\)
\(L(\frac12)\) \(\approx\) \(1.61318 + 0.237028i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (2.12 - 0.707i)T \)
good7 \( 1 + (1.44 + 1.44i)T + 7iT^{2} \)
11 \( 1 - 4.87T + 11T^{2} \)
13 \( 1 + (-4.87 + 4.87i)T - 13iT^{2} \)
17 \( 1 + (2 - 2i)T - 17iT^{2} \)
19 \( 1 - 0.635iT - 19T^{2} \)
23 \( 1 + (1.55 - 1.55i)T - 23iT^{2} \)
29 \( 1 - 4.24T + 29T^{2} \)
31 \( 1 - 6.89iT - 31T^{2} \)
37 \( 1 + (-1.41 - 1.41i)T + 37iT^{2} \)
41 \( 1 - 10.8T + 41T^{2} \)
43 \( 1 + (-3.46 - 3.46i)T + 43iT^{2} \)
47 \( 1 + (-4.44 - 4.44i)T + 47iT^{2} \)
53 \( 1 + (-5.51 + 5.51i)T - 53iT^{2} \)
59 \( 1 - 13.3iT - 59T^{2} \)
61 \( 1 + 4.09iT - 61T^{2} \)
67 \( 1 + (-9.75 + 9.75i)T - 67iT^{2} \)
71 \( 1 - 0.898iT - 71T^{2} \)
73 \( 1 + (10.7 + 10.7i)T + 73iT^{2} \)
79 \( 1 + 10.8T + 79T^{2} \)
83 \( 1 + (2.04 + 2.04i)T + 83iT^{2} \)
89 \( 1 + 5.10iT - 89T^{2} \)
97 \( 1 + (5 - 5i)T - 97iT^{2} \)
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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

−10.23038156278308723703973375961, −9.089914420003826778186144873253, −8.465851806628220855041615884612, −7.64162439414097737071946864603, −6.68271881515380986182637977276, −5.92072348298939862851281768810, −4.35850604415172941133633021891, −3.74998368107670656516784290196, −3.03863116668144141358883846635, −1.04642416016545757615662787297, 1.04799140487327398111080957224, 2.49185766853039012594664897586, 3.90207602176791134716403756496, 4.22848005472224011290924727767, 5.95178123786861788987871698195, 6.63292873388374814954855623727, 7.39704375782683807714511850920, 8.660562761863171787290106694260, 8.871116913763711054179066631637, 9.653510022934340400464556661338

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