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Properties

Label 2-960-16.5-c3-0-39
Degree $2$
Conductor $960$
Sign $-0.605 + 0.796i$
Analytic cond. $56.6418$
Root an. cond. $7.52607$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.12 − 2.12i)3-s + (−3.53 − 3.53i)5-s − 16.8i·7-s − 8.99i·9-s + (5.27 + 5.27i)11-s + (17.9 − 17.9i)13-s − 15·15-s + 98.3·17-s + (−4.92 + 4.92i)19-s + (−35.7 − 35.7i)21-s − 13.7i·23-s + 25.0i·25-s + (−19.0 − 19.0i)27-s + (198. − 198. i)29-s + 6.63·31-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (−0.316 − 0.316i)5-s − 0.910i·7-s − 0.333i·9-s + (0.144 + 0.144i)11-s + (0.383 − 0.383i)13-s − 0.258·15-s + 1.40·17-s + (−0.0594 + 0.0594i)19-s + (−0.371 − 0.371i)21-s − 0.125i·23-s + 0.200i·25-s + (−0.136 − 0.136i)27-s + (1.27 − 1.27i)29-s + 0.0384·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.796i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.605 + 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $-0.605 + 0.796i$
Analytic conductor: \(56.6418\)
Root analytic conductor: \(7.52607\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :3/2),\ -0.605 + 0.796i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.076231764\)
\(L(\frac12)\) \(\approx\) \(2.076231764\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.12 + 2.12i)T \)
5 \( 1 + (3.53 + 3.53i)T \)
good7 \( 1 + 16.8iT - 343T^{2} \)
11 \( 1 + (-5.27 - 5.27i)T + 1.33e3iT^{2} \)
13 \( 1 + (-17.9 + 17.9i)T - 2.19e3iT^{2} \)
17 \( 1 - 98.3T + 4.91e3T^{2} \)
19 \( 1 + (4.92 - 4.92i)T - 6.85e3iT^{2} \)
23 \( 1 + 13.7iT - 1.21e4T^{2} \)
29 \( 1 + (-198. + 198. i)T - 2.43e4iT^{2} \)
31 \( 1 - 6.63T + 2.97e4T^{2} \)
37 \( 1 + (-41.4 - 41.4i)T + 5.06e4iT^{2} \)
41 \( 1 + 35.5iT - 6.89e4T^{2} \)
43 \( 1 + (28.8 + 28.8i)T + 7.95e4iT^{2} \)
47 \( 1 + 461.T + 1.03e5T^{2} \)
53 \( 1 + (139. + 139. i)T + 1.48e5iT^{2} \)
59 \( 1 + (412. + 412. i)T + 2.05e5iT^{2} \)
61 \( 1 + (175. - 175. i)T - 2.26e5iT^{2} \)
67 \( 1 + (299. - 299. i)T - 3.00e5iT^{2} \)
71 \( 1 + 1.06e3iT - 3.57e5T^{2} \)
73 \( 1 + 489. iT - 3.89e5T^{2} \)
79 \( 1 - 559.T + 4.93e5T^{2} \)
83 \( 1 + (590. - 590. i)T - 5.71e5iT^{2} \)
89 \( 1 - 1.59e3iT - 7.04e5T^{2} \)
97 \( 1 + 154.T + 9.12e5T^{2} \)
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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

−9.401485161605866692162788638340, −8.154824077440225113780920560336, −7.921076101800205152903119497959, −6.90345040351703087072717611378, −6.02519902364587058474306405061, −4.82714971482050323167436627097, −3.86090473746822457642496844773, −2.98459370458019204269385343914, −1.46752507636805182735448951658, −0.53406427716429212344753213386, 1.37269021801991583101994432920, 2.77217514558133340337536523019, 3.47128960146875375392313516387, 4.63247802905336503099416347055, 5.58826751184600953574336215939, 6.49551820732802525493576937257, 7.55422292910607541149516526553, 8.402080673931894909621591535391, 9.029616499155110748480762139484, 9.901577371195472886243792711195

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