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Properties

Label 2-960-24.11-c1-0-18
Degree $2$
Conductor $960$
Sign $0.985 + 0.169i$
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  + (1.41 − i)3-s + 5-s + 2i·7-s + (1.00 − 2.82i)9-s + 2i·11-s + 2.82i·13-s + (1.41 − i)15-s − 2.82i·17-s + 5.65·19-s + (2 + 2.82i)21-s + 8.48·23-s + 25-s + (−1.41 − 5.00i)27-s − 2·29-s + 6i·31-s + ⋯
L(s)  = 1  + (0.816 − 0.577i)3-s + 0.447·5-s + 0.755i·7-s + (0.333 − 0.942i)9-s + 0.603i·11-s + 0.784i·13-s + (0.365 − 0.258i)15-s − 0.685i·17-s + 1.29·19-s + (0.436 + 0.617i)21-s + 1.76·23-s + 0.200·25-s + (−0.272 − 0.962i)27-s − 0.371·29-s + 1.07i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $0.985 + 0.169i$
Analytic conductor: \(7.66563\)
Root analytic conductor: \(2.76868\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (671, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1/2),\ 0.985 + 0.169i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.36387 - 0.201317i\)
\(L(\frac12)\) \(\approx\) \(2.36387 - 0.201317i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.41 + i)T \)
5 \( 1 - T \)
good7 \( 1 - 2iT - 7T^{2} \)
11 \( 1 - 2iT - 11T^{2} \)
13 \( 1 - 2.82iT - 13T^{2} \)
17 \( 1 + 2.82iT - 17T^{2} \)
19 \( 1 - 5.65T + 19T^{2} \)
23 \( 1 - 8.48T + 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 - 6iT - 31T^{2} \)
37 \( 1 + 2.82iT - 37T^{2} \)
41 \( 1 + 11.3iT - 41T^{2} \)
43 \( 1 + 8.48T + 43T^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 + 10T + 53T^{2} \)
59 \( 1 - 10iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 2.82T + 67T^{2} \)
71 \( 1 + 5.65T + 71T^{2} \)
73 \( 1 - 10T + 73T^{2} \)
79 \( 1 + 2iT - 79T^{2} \)
83 \( 1 + 2iT - 83T^{2} \)
89 \( 1 + 5.65iT - 89T^{2} \)
97 \( 1 + 14T + 97T^{2} \)
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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

−9.583163593809097437299550599020, −9.213699071641436836996106608254, −8.522343860292085741151064288014, −7.21596265614789437057392845132, −6.98577340369585245011517830498, −5.68627750473301771784588732821, −4.79649688976664199970180325729, −3.37000754329997649828098296909, −2.48108612847455312881335281237, −1.40816925304107073261244093385, 1.27478694381970216509052049936, 2.88159624100861602716160471689, 3.53868158663068580108511800570, 4.73114161479892060249917934667, 5.55446343215885711025605926362, 6.75317446075505545598636591550, 7.75401416109773137137303057801, 8.334580893186429159596594113267, 9.413631176483512448572350101949, 9.864286539450045519397581675687

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