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Properties

Label 2-960-80.3-c1-0-0
Degree $2$
Conductor $960$
Sign $-0.675 - 0.737i$
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 − 2i)5-s + (−2.10 + 2.10i)7-s + 9-s + (−2.10 − 2.10i)11-s + (−1 + 2i)15-s + (−4.62 + 4.62i)17-s + (3.52 + 3.52i)19-s + (2.10 − 2.10i)21-s + (−3.52 − 3.52i)23-s + (−3 − 4i)25-s − 27-s + (−1 + i)29-s + 4.20i·31-s + (2.10 + 2.10i)33-s + ⋯
L(s)  = 1  − 0.577·3-s + (0.447 − 0.894i)5-s + (−0.794 + 0.794i)7-s + 0.333·9-s + (−0.634 − 0.634i)11-s + (−0.258 + 0.516i)15-s + (−1.12 + 1.12i)17-s + (0.808 + 0.808i)19-s + (0.458 − 0.458i)21-s + (−0.734 − 0.734i)23-s + (−0.600 − 0.800i)25-s − 0.192·27-s + (−0.185 + 0.185i)29-s + 0.755i·31-s + (0.366 + 0.366i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.675 - 0.737i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.675 - 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $-0.675 - 0.737i$
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: \((2,\ 960,\ (\ :1/2),\ -0.675 - 0.737i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.145476 + 0.330476i\)
\(L(\frac12)\) \(\approx\) \(0.145476 + 0.330476i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 + 2i)T \)
good7 \( 1 + (2.10 - 2.10i)T - 7iT^{2} \)
11 \( 1 + (2.10 + 2.10i)T + 11iT^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + (4.62 - 4.62i)T - 17iT^{2} \)
19 \( 1 + (-3.52 - 3.52i)T + 19iT^{2} \)
23 \( 1 + (3.52 + 3.52i)T + 23iT^{2} \)
29 \( 1 + (1 - i)T - 29iT^{2} \)
31 \( 1 - 4.20iT - 31T^{2} \)
37 \( 1 - 7.25iT - 37T^{2} \)
41 \( 1 - 4iT - 41T^{2} \)
43 \( 1 - 3.04iT - 43T^{2} \)
47 \( 1 + (4.68 + 4.68i)T + 47iT^{2} \)
53 \( 1 + 3.15T + 53T^{2} \)
59 \( 1 + (5.15 - 5.15i)T - 59iT^{2} \)
61 \( 1 + (6.62 + 6.62i)T + 61iT^{2} \)
67 \( 1 - 7.45iT - 67T^{2} \)
71 \( 1 - 12.4T + 71T^{2} \)
73 \( 1 + (10.2 - 10.2i)T - 73iT^{2} \)
79 \( 1 + 12.4T + 79T^{2} \)
83 \( 1 - 16.4T + 83T^{2} \)
89 \( 1 + 13.2T + 89T^{2} \)
97 \( 1 + (11.4 - 11.4i)T - 97iT^{2} \)
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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

−10.21313313487875720040049170718, −9.593239977004353411998925443630, −8.614352948312992263764564275251, −8.107680892488276187614337269941, −6.58881144860437999507040209684, −5.98888478179368792824150640221, −5.30146185428900166165531885841, −4.27226115505485552661359944671, −2.94341098669229246157888702466, −1.58832891180388583091302789527, 0.17220617783880035075704611158, 2.16213065691106612449605826504, 3.25632169758669597652416783012, 4.41487055887577928336258611264, 5.45726182488539589078867370129, 6.40119216135070296935945260637, 7.16640345543534225853882174018, 7.59149908578619047819911755029, 9.366747040675919551359336315155, 9.684351520254041689327825727997

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