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Properties

Label 2-960-80.3-c1-0-2
Degree $2$
Conductor $960$
Sign $-0.642 - 0.766i$
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  + 3-s + (−1.76 + 1.37i)5-s + (−0.159 + 0.159i)7-s + 9-s + (−1.60 − 1.60i)11-s + 4.36i·13-s + (−1.76 + 1.37i)15-s + (−4.63 + 4.63i)17-s + (−3.97 − 3.97i)19-s + (−0.159 + 0.159i)21-s + (5.58 + 5.58i)23-s + (1.20 − 4.85i)25-s + 27-s + (−6.25 + 6.25i)29-s + 1.69i·31-s + ⋯
L(s)  = 1  + 0.577·3-s + (−0.787 + 0.615i)5-s + (−0.0602 + 0.0602i)7-s + 0.333·9-s + (−0.485 − 0.485i)11-s + 1.21i·13-s + (−0.454 + 0.355i)15-s + (−1.12 + 1.12i)17-s + (−0.912 − 0.912i)19-s + (−0.0348 + 0.0348i)21-s + (1.16 + 1.16i)23-s + (0.241 − 0.970i)25-s + 0.192·27-s + (−1.16 + 1.16i)29-s + 0.304i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $-0.642 - 0.766i$
Analytic conductor: \(7.66563\)
Root analytic conductor: \(2.76868\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (943, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1/2),\ -0.642 - 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.399405 + 0.855527i\)
\(L(\frac12)\) \(\approx\) \(0.399405 + 0.855527i\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
5 \( 1 + (1.76 - 1.37i)T \)
good7 \( 1 + (0.159 - 0.159i)T - 7iT^{2} \)
11 \( 1 + (1.60 + 1.60i)T + 11iT^{2} \)
13 \( 1 - 4.36iT - 13T^{2} \)
17 \( 1 + (4.63 - 4.63i)T - 17iT^{2} \)
19 \( 1 + (3.97 + 3.97i)T + 19iT^{2} \)
23 \( 1 + (-5.58 - 5.58i)T + 23iT^{2} \)
29 \( 1 + (6.25 - 6.25i)T - 29iT^{2} \)
31 \( 1 - 1.69iT - 31T^{2} \)
37 \( 1 + 0.609iT - 37T^{2} \)
41 \( 1 + 0.538iT - 41T^{2} \)
43 \( 1 - 0.592iT - 43T^{2} \)
47 \( 1 + (4.85 + 4.85i)T + 47iT^{2} \)
53 \( 1 + 4.82T + 53T^{2} \)
59 \( 1 + (5.78 - 5.78i)T - 59iT^{2} \)
61 \( 1 + (-1.65 - 1.65i)T + 61iT^{2} \)
67 \( 1 + 0.485iT - 67T^{2} \)
71 \( 1 + 6.86T + 71T^{2} \)
73 \( 1 + (0.160 - 0.160i)T - 73iT^{2} \)
79 \( 1 - 7.13T + 79T^{2} \)
83 \( 1 - 6.88T + 83T^{2} \)
89 \( 1 - 17.0T + 89T^{2} \)
97 \( 1 + (-9.64 + 9.64i)T - 97iT^{2} \)
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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

−10.59227111779801904484529424600, −9.124717627062371745266067031636, −8.849614166997807463237344433623, −7.78656564883333024077079899226, −7.00302677874579309679780763161, −6.32672843483025581737668478659, −4.87083601656267944645155070362, −3.94629776699935727464161081488, −3.07956788137075049586958530804, −1.89767656032385830369954035936, 0.38799710493941765343744282812, 2.20809821219807488696474788352, 3.31812055254389359988814680677, 4.41421757033410371114788356224, 5.08637640564920940297480722741, 6.40939899944762547147945041087, 7.48803344664947339719635294789, 8.026849043374878960292966249982, 8.809686496887360086012748695317, 9.595958443687811904121447405888

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