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Properties

Label 2-1040-13.4-c1-0-2
Degree $2$
Conductor $1040$
Sign $-0.993 - 0.111i$
Analytic cond. $8.30444$
Root an. cond. $2.88174$
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 + 2.44i)3-s − i·5-s + (1.64 − 0.952i)7-s + (−2.49 − 4.32i)9-s + (−0.926 − 0.534i)11-s + (1.40 + 3.32i)13-s + (2.44 + 1.41i)15-s + (0.318 + 0.551i)17-s + (−4.96 + 2.86i)19-s + 5.38i·21-s + (−1.90 + 3.30i)23-s − 25-s + 5.62·27-s + (−4.72 + 8.18i)29-s + 1.46i·31-s + ⋯
L(s)  = 1  + (−0.816 + 1.41i)3-s − 0.447i·5-s + (0.623 − 0.360i)7-s + (−0.831 − 1.44i)9-s + (−0.279 − 0.161i)11-s + (0.388 + 0.921i)13-s + (0.632 + 0.364i)15-s + (0.0772 + 0.133i)17-s + (−1.13 + 0.657i)19-s + 1.17i·21-s + (−0.398 + 0.689i)23-s − 0.200·25-s + 1.08·27-s + (−0.877 + 1.52i)29-s + 0.262i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.111i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 - 0.111i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1040\)    =    \(2^{4} \cdot 5 \cdot 13\)
Sign: $-0.993 - 0.111i$
Analytic conductor: \(8.30444\)
Root analytic conductor: \(2.88174\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1040} (641, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1040,\ (\ :1/2),\ -0.993 - 0.111i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6488200348\)
\(L(\frac12)\) \(\approx\) \(0.6488200348\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + iT \)
13 \( 1 + (-1.40 - 3.32i)T \)
good3 \( 1 + (1.41 - 2.44i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + (-1.64 + 0.952i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.926 + 0.534i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.318 - 0.551i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.96 - 2.86i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.90 - 3.30i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.72 - 8.18i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.46iT - 31T^{2} \)
37 \( 1 + (-0.655 - 0.378i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.232 + 0.133i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.318 + 0.551i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 9.44iT - 47T^{2} \)
53 \( 1 + 6.99T + 53T^{2} \)
59 \( 1 + (-0.641 + 0.370i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.09 + 3.63i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.01 - 4.04i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (8.45 - 4.88i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 3.71iT - 73T^{2} \)
79 \( 1 - 9.31T + 79T^{2} \)
83 \( 1 + 5.11iT - 83T^{2} \)
89 \( 1 + (10.8 + 6.28i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.65 - 2.11i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.46108825991921546403648530090, −9.587760122583437137704282420920, −8.863805499635448990670330484298, −8.038558078702292143081777672017, −6.78689771940226227223934576895, −5.76832908331126913422684790837, −5.07320049351003945936416450616, −4.24878162433682793593426295846, −3.60645866163197157502938221720, −1.65169809704248167040615075568, 0.31981793992461268694053023207, 1.81499067273240447901866152800, 2.71282136367170379247637195132, 4.39300526686235526451551561084, 5.54170899896583912972814779773, 6.15461014763035364272316090624, 6.91874934137313711451230009929, 7.904994886314761012324824847880, 8.220223649523760736025751240815, 9.571184189274704014956333157988

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