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Properties

Label 2-960-15.2-c1-0-33
Degree $2$
Conductor $960$
Sign $-0.920 + 0.391i$
Analytic cond. $7.66563$
Root an. cond. $2.76868$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + 1.41i)3-s + (−1 − 2i)5-s + (0.414 + 0.414i)7-s + (−1.00 − 2.82i)9-s + 4.82i·11-s + (1.82 − 1.82i)13-s + (3.82 + 0.585i)15-s + (−3.82 + 3.82i)17-s − 4.82i·19-s + (−1 + 0.171i)21-s + (−1.58 − 1.58i)23-s + (−3 + 4i)25-s + (5.00 + 1.41i)27-s − 7.65·29-s − 5.65·31-s + ⋯
L(s)  = 1  + (−0.577 + 0.816i)3-s + (−0.447 − 0.894i)5-s + (0.156 + 0.156i)7-s + (−0.333 − 0.942i)9-s + 1.45i·11-s + (0.507 − 0.507i)13-s + (0.988 + 0.151i)15-s + (−0.928 + 0.928i)17-s − 1.10i·19-s + (−0.218 + 0.0374i)21-s + (−0.330 − 0.330i)23-s + (−0.600 + 0.800i)25-s + (0.962 + 0.272i)27-s − 1.42·29-s − 1.01·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.920 + 0.391i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.920 + 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $-0.920 + 0.391i$
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: \(1\)
Selberg data: \((2,\ 960,\ (\ :1/2),\ -0.920 + 0.391i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.414 - 0.414i)T + 7iT^{2} \)
11 \( 1 - 4.82iT - 11T^{2} \)
13 \( 1 + (-1.82 + 1.82i)T - 13iT^{2} \)
17 \( 1 + (3.82 - 3.82i)T - 17iT^{2} \)
19 \( 1 + 4.82iT - 19T^{2} \)
23 \( 1 + (1.58 + 1.58i)T + 23iT^{2} \)
29 \( 1 + 7.65T + 29T^{2} \)
31 \( 1 + 5.65T + 31T^{2} \)
37 \( 1 + (-0.171 - 0.171i)T + 37iT^{2} \)
41 \( 1 + 5.65iT - 41T^{2} \)
43 \( 1 + (2.41 - 2.41i)T - 43iT^{2} \)
47 \( 1 + (6.41 - 6.41i)T - 47iT^{2} \)
53 \( 1 + (3 + 3i)T + 53iT^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 + 11.6T + 61T^{2} \)
67 \( 1 + (-4.07 - 4.07i)T + 67iT^{2} \)
71 \( 1 + 6.48iT - 71T^{2} \)
73 \( 1 + (-6.65 + 6.65i)T - 73iT^{2} \)
79 \( 1 + 4.82iT - 79T^{2} \)
83 \( 1 + (5.24 + 5.24i)T + 83iT^{2} \)
89 \( 1 + 4.34T + 89T^{2} \)
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

−9.511995756611773823661983366507, −9.057212057446693284497429386034, −8.128883605939107729915570693068, −7.09861619378050080189685092234, −6.04517976364497022416598265293, −5.05597908403725179781238658724, −4.46746354144859154210867854418, −3.60237164014713376360508706984, −1.83523356118171757667998848040, 0, 1.70408717219786167580353051772, 3.04848167123032299079881373222, 4.05646868522523180975860633220, 5.47947161797779997019372768288, 6.20757758826061290540124485319, 6.96169799701934089203530988463, 7.77253550790715272733027374530, 8.472273039498786043944708504414, 9.590952190477366209471433372972

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