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Properties

Label 2-1040-13.12-c3-0-1
Degree $2$
Conductor $1040$
Sign $-0.994 + 0.108i$
Analytic cond. $61.3619$
Root an. cond. $7.83338$
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  − 5.13·3-s − 5i·5-s + 9.97i·7-s − 0.594·9-s + 6.66i·11-s + (5.10 + 46.5i)13-s + 25.6i·15-s − 1.15·17-s − 48.3i·19-s − 51.2i·21-s + 125.·23-s − 25·25-s + 141.·27-s − 139.·29-s + 195. i·31-s + ⋯
L(s)  = 1  − 0.988·3-s − 0.447i·5-s + 0.538i·7-s − 0.0220·9-s + 0.182i·11-s + (0.108 + 0.994i)13-s + 0.442i·15-s − 0.0164·17-s − 0.583i·19-s − 0.532i·21-s + 1.13·23-s − 0.200·25-s + 1.01·27-s − 0.890·29-s + 1.13i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.108i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.994 + 0.108i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1040\)    =    \(2^{4} \cdot 5 \cdot 13\)
Sign: $-0.994 + 0.108i$
Analytic conductor: \(61.3619\)
Root analytic conductor: \(7.83338\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1040} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1040,\ (\ :3/2),\ -0.994 + 0.108i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.09021361210\)
\(L(\frac12)\) \(\approx\) \(0.09021361210\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + 5iT \)
13 \( 1 + (-5.10 - 46.5i)T \)
good3 \( 1 + 5.13T + 27T^{2} \)
7 \( 1 - 9.97iT - 343T^{2} \)
11 \( 1 - 6.66iT - 1.33e3T^{2} \)
17 \( 1 + 1.15T + 4.91e3T^{2} \)
19 \( 1 + 48.3iT - 6.85e3T^{2} \)
23 \( 1 - 125.T + 1.21e4T^{2} \)
29 \( 1 + 139.T + 2.43e4T^{2} \)
31 \( 1 - 195. iT - 2.97e4T^{2} \)
37 \( 1 - 174. iT - 5.06e4T^{2} \)
41 \( 1 + 452. iT - 6.89e4T^{2} \)
43 \( 1 + 54.5T + 7.95e4T^{2} \)
47 \( 1 + 369. iT - 1.03e5T^{2} \)
53 \( 1 - 4.19T + 1.48e5T^{2} \)
59 \( 1 - 169. iT - 2.05e5T^{2} \)
61 \( 1 - 271.T + 2.26e5T^{2} \)
67 \( 1 - 54.6iT - 3.00e5T^{2} \)
71 \( 1 - 883. iT - 3.57e5T^{2} \)
73 \( 1 - 613. iT - 3.89e5T^{2} \)
79 \( 1 + 425.T + 4.93e5T^{2} \)
83 \( 1 + 680. iT - 5.71e5T^{2} \)
89 \( 1 - 943. iT - 7.04e5T^{2} \)
97 \( 1 + 1.81e3iT - 9.12e5T^{2} \)
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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

−10.03794663937886763845696871023, −8.932859262983849423366748389547, −8.671720774689602817344996007144, −7.18694332294426882560758149059, −6.60688903000381187530312981862, −5.49273365433453969567867563357, −5.07858107652752673382288449636, −3.96708897698901715877327539930, −2.57267102740242491521633203316, −1.27668295972434811186816726460, 0.03052849053066024677894357322, 1.09785856010436159036333084810, 2.73538134897465057982263769395, 3.74855910923941880614780616842, 4.91676263575623716804555379181, 5.76807496433113703537107471222, 6.39213956545287786202535579879, 7.41055526336071297495219401113, 8.094376261377264329116438801861, 9.266059179202406987444637160610

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