Properties

Label 2-3040-760.189-c0-0-6
Degree $2$
Conductor $3040$
Sign $-0.382 + 0.923i$
Analytic cond. $1.51715$
Root an. cond. $1.23172$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.765i·3-s − i·5-s + 0.414·9-s + 1.41i·11-s − 1.84i·13-s − 0.765·15-s − i·19-s − 25-s − 1.08i·27-s + 1.08·33-s + 0.765i·37-s − 1.41·39-s − 0.414i·45-s + 49-s − 0.765i·53-s + ⋯
L(s)  = 1  − 0.765i·3-s − i·5-s + 0.414·9-s + 1.41i·11-s − 1.84i·13-s − 0.765·15-s − i·19-s − 25-s − 1.08i·27-s + 1.08·33-s + 0.765i·37-s − 1.41·39-s − 0.414i·45-s + 49-s − 0.765i·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3040\)    =    \(2^{5} \cdot 5 \cdot 19\)
Sign: $-0.382 + 0.923i$
Analytic conductor: \(1.51715\)
Root analytic conductor: \(1.23172\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3040} (1329, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3040,\ (\ :0),\ -0.382 + 0.923i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.232569570\)
\(L(\frac12)\) \(\approx\) \(1.232569570\)
\(L(1)\) 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 \( 1 + iT \)
19 \( 1 + iT \)
good3 \( 1 + 0.765iT - T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 - 1.41iT - T^{2} \)
13 \( 1 + 1.84iT - T^{2} \)
17 \( 1 - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - 0.765iT - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + 0.765iT - T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - 1.41iT - T^{2} \)
67 \( 1 + 1.84iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + 1.84T + T^{2} \)
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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

−8.510562077482245348000165747557, −7.82653642611360885386788181503, −7.31764051427316498990084897358, −6.52211761418232171119494692274, −5.50334148124619558267464971558, −4.87007511038573327979103700394, −4.10468502200581449912016588106, −2.80588263874754869932700364263, −1.81332262172367766866011853704, −0.797008298460014724923355452999, 1.64141666770680814925516345457, 2.79264287469240513110144976115, 3.84444300515303348981761995708, 4.09940865972345904424854948415, 5.36102663002055108871756388929, 6.15229139603184897096641117929, 6.82151305814729224620667991683, 7.55617152355676713527510408460, 8.523264658287442963850600149622, 9.233036733300187439417239829514

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