Properties

Label 2-960-15.2-c1-0-11
Degree $2$
Conductor $960$
Sign $-0.0618 - 0.998i$
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  + (1 + 1.41i)3-s + (−1 − 2i)5-s + (2.41 + 2.41i)7-s + (−1.00 + 2.82i)9-s + 0.828i·11-s + (−3.82 + 3.82i)13-s + (1.82 − 3.41i)15-s + (1.82 − 1.82i)17-s − 0.828i·19-s + (−1 + 5.82i)21-s + (4.41 + 4.41i)23-s + (−3 + 4i)25-s + (−5.00 + 1.41i)27-s + 3.65·29-s − 5.65·31-s + ⋯
L(s)  = 1  + (0.577 + 0.816i)3-s + (−0.447 − 0.894i)5-s + (0.912 + 0.912i)7-s + (−0.333 + 0.942i)9-s + 0.249i·11-s + (−1.06 + 1.06i)13-s + (0.472 − 0.881i)15-s + (0.443 − 0.443i)17-s − 0.190i·19-s + (−0.218 + 1.27i)21-s + (0.920 + 0.920i)23-s + (−0.600 + 0.800i)25-s + (−0.962 + 0.272i)27-s + 0.679·29-s − 1.01·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0618 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0618 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $-0.0618 - 0.998i$
Analytic conductor: \(7.66563\)
Root analytic conductor: \(2.76868\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1/2),\ -0.0618 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18395 + 1.25961i\)
\(L(\frac12)\) \(\approx\) \(1.18395 + 1.25961i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 - 1.41i)T \)
5 \( 1 + (1 + 2i)T \)
good7 \( 1 + (-2.41 - 2.41i)T + 7iT^{2} \)
11 \( 1 - 0.828iT - 11T^{2} \)
13 \( 1 + (3.82 - 3.82i)T - 13iT^{2} \)
17 \( 1 + (-1.82 + 1.82i)T - 17iT^{2} \)
19 \( 1 + 0.828iT - 19T^{2} \)
23 \( 1 + (-4.41 - 4.41i)T + 23iT^{2} \)
29 \( 1 - 3.65T + 29T^{2} \)
31 \( 1 + 5.65T + 31T^{2} \)
37 \( 1 + (-5.82 - 5.82i)T + 37iT^{2} \)
41 \( 1 - 5.65iT - 41T^{2} \)
43 \( 1 + (0.414 - 0.414i)T - 43iT^{2} \)
47 \( 1 + (-3.58 + 3.58i)T - 47iT^{2} \)
53 \( 1 + (3 + 3i)T + 53iT^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 + 0.343T + 61T^{2} \)
67 \( 1 + (-10.0 - 10.0i)T + 67iT^{2} \)
71 \( 1 + 10.4iT - 71T^{2} \)
73 \( 1 + (4.65 - 4.65i)T - 73iT^{2} \)
79 \( 1 + 0.828iT - 79T^{2} \)
83 \( 1 + (3.24 + 3.24i)T + 83iT^{2} \)
89 \( 1 + 15.6T + 89T^{2} \)
97 \( 1 + (-1 - i)T + 97iT^{2} \)
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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

−9.897355350016418063688651535729, −9.348770888244808371778173476794, −8.690765939558606359137909216549, −7.965161892173020048476999045748, −7.13360602323748857442268142954, −5.44132200062423896186386957424, −4.92476569234200957237703195770, −4.22412629332306030864663358076, −2.87857729113508230982392466025, −1.72808411763103376484315747461, 0.77169070445629560534067818795, 2.31965239047361894799867454546, 3.26321864936342850208640839902, 4.28044205259384653661742286434, 5.57128096838569768790162557447, 6.70904781087328332561528758163, 7.51443484413690905683471163414, 7.80941496583141341812565393176, 8.718094599964684680209058109410, 9.945205556486058131327776376520

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