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Properties

Label 2-960-12.11-c1-0-26
Degree $2$
Conductor $960$
Sign $-0.577 + 0.816i$
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 + i·5-s − 0.828i·7-s + (−1.00 − 2.82i)9-s − 0.828·11-s − 6.82·13-s + (1.41 + i)15-s − 4.82i·17-s − 6i·19-s + (−1.17 − 0.828i)21-s + 4.82·23-s − 25-s + (−5.00 − 1.41i)27-s − 6i·29-s − 2i·31-s + ⋯
L(s)  = 1  + (0.577 − 0.816i)3-s + 0.447i·5-s − 0.313i·7-s + (−0.333 − 0.942i)9-s − 0.249·11-s − 1.89·13-s + (0.365 + 0.258i)15-s − 1.17i·17-s − 1.37i·19-s + (−0.255 − 0.180i)21-s + 1.00·23-s − 0.200·25-s + (−0.962 − 0.272i)27-s − 1.11i·29-s − 0.359i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $-0.577 + 0.816i$
Analytic conductor: \(7.66563\)
Root analytic conductor: \(2.76868\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1/2),\ -0.577 + 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.623102 - 1.20374i\)
\(L(\frac12)\) \(\approx\) \(0.623102 - 1.20374i\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
good7 \( 1 + 0.828iT - 7T^{2} \)
11 \( 1 + 0.828T + 11T^{2} \)
13 \( 1 + 6.82T + 13T^{2} \)
17 \( 1 + 4.82iT - 17T^{2} \)
19 \( 1 + 6iT - 19T^{2} \)
23 \( 1 - 4.82T + 23T^{2} \)
29 \( 1 + 6iT - 29T^{2} \)
31 \( 1 + 2iT - 31T^{2} \)
37 \( 1 + 1.17T + 37T^{2} \)
41 \( 1 + 1.65iT - 41T^{2} \)
43 \( 1 - 6.82iT - 43T^{2} \)
47 \( 1 - 8.82T + 47T^{2} \)
53 \( 1 + 3.65iT - 53T^{2} \)
59 \( 1 - 7.17T + 59T^{2} \)
61 \( 1 + 9.31T + 61T^{2} \)
67 \( 1 - 12.4iT - 67T^{2} \)
71 \( 1 + 11.3T + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 + 6iT - 79T^{2} \)
83 \( 1 - 17.3T + 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + 6T + 97T^{2} \)
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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

−9.543072496241233007726291639856, −9.031623997655702065616084320319, −7.73909224668552255437444309810, −7.27918622735936250058180809601, −6.69637394051890865558779125688, −5.37919165809038594767477739801, −4.38832025763114647239075010323, −2.85805945065386545787723181419, −2.43507371833463527432156924322, −0.55531404142086397403715491696, 1.91537324480910516310394356639, 3.02829383146172825813076352534, 4.11443753035682927731730057662, 5.03677756902174926644935968214, 5.69194332417270120411632559774, 7.16125397011257398292963863467, 7.956003334789838103123225040575, 8.780976908028003295142327559595, 9.420820900062018258890477373633, 10.31634410407111899985739548471

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