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Properties

Label 1-3040-3040.643-r0-0-0
Degree $1$
Conductor $3040$
Sign $-0.223 + 0.974i$
Analytic cond. $14.1177$
Root an. cond. $14.1177$
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  + (0.0871 − 0.996i)3-s + (−0.5 + 0.866i)7-s + (−0.984 − 0.173i)9-s + (0.965 + 0.258i)11-s + (0.996 − 0.0871i)13-s + (−0.984 + 0.173i)17-s + (0.819 + 0.573i)21-s + (−0.939 + 0.342i)23-s + (−0.258 + 0.965i)27-s + (−0.819 + 0.573i)29-s + (0.5 − 0.866i)31-s + (0.342 − 0.939i)33-s + (0.707 + 0.707i)37-s − i·39-s + (0.642 + 0.766i)41-s + ⋯
L(s)  = 1  + (0.0871 − 0.996i)3-s + (−0.5 + 0.866i)7-s + (−0.984 − 0.173i)9-s + (0.965 + 0.258i)11-s + (0.996 − 0.0871i)13-s + (−0.984 + 0.173i)17-s + (0.819 + 0.573i)21-s + (−0.939 + 0.342i)23-s + (−0.258 + 0.965i)27-s + (−0.819 + 0.573i)29-s + (0.5 − 0.866i)31-s + (0.342 − 0.939i)33-s + (0.707 + 0.707i)37-s − i·39-s + (0.642 + 0.766i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.223 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.223 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3040\)    =    \(2^{5} \cdot 5 \cdot 19\)
Sign: $-0.223 + 0.974i$
Analytic conductor: \(14.1177\)
Root analytic conductor: \(14.1177\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3040} (643, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3040,\ (0:\ ),\ -0.223 + 0.974i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4086690121 + 0.5128547381i\)
\(L(\frac12)\) \(\approx\) \(0.4086690121 + 0.5128547381i\)
\(L(1)\) \(\approx\) \(0.8902832069 - 0.09063731651i\)
\(L(1)\) \(\approx\) \(0.8902832069 - 0.09063731651i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
19 \( 1 \)
good3 \( 1 + (0.0871 - 0.996i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (0.965 + 0.258i)T \)
13 \( 1 + (0.996 - 0.0871i)T \)
17 \( 1 + (-0.984 + 0.173i)T \)
23 \( 1 + (-0.939 + 0.342i)T \)
29 \( 1 + (-0.819 + 0.573i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (0.707 + 0.707i)T \)
41 \( 1 + (0.642 + 0.766i)T \)
43 \( 1 + (0.422 + 0.906i)T \)
47 \( 1 + (-0.984 - 0.173i)T \)
53 \( 1 + (-0.906 - 0.422i)T \)
59 \( 1 + (-0.819 - 0.573i)T \)
61 \( 1 + (-0.906 - 0.422i)T \)
67 \( 1 + (0.573 + 0.819i)T \)
71 \( 1 + (0.342 - 0.939i)T \)
73 \( 1 + (-0.766 + 0.642i)T \)
79 \( 1 + (-0.766 + 0.642i)T \)
83 \( 1 + (-0.258 - 0.965i)T \)
89 \( 1 + (-0.642 + 0.766i)T \)
97 \( 1 + (0.984 - 0.173i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.00647391122273259495415864976, −17.99330920180998285056054221267, −17.26457999606608485769476664727, −16.6871366080949453007379426690, −15.96490247849848251171299549725, −15.61430591337817936470707142528, −14.5569616605565148147657585491, −13.9673993641523059843992811420, −13.48106413927540430882268471746, −12.48056954139149227993205269282, −11.519094775124606445646576903472, −10.94693280717492455632463068698, −10.38543345489531399569050576555, −9.48327925136491418475177276210, −9.020496291039903081097307605091, −8.2470944007706273703620474191, −7.27688634986780626728947734520, −6.30101367845858409720246637380, −5.92250637409471488712084732406, −4.64318845128057329612814885559, −4.040561383701794430226032535739, −3.59020525483867806687271745747, −2.605984045070735882567437978811, −1.42874030328567422666648430706, −0.195377879399534784743363215912, 1.25827738013980709189633675578, 1.91595772079732548729495777677, 2.81022202716281485772414217939, 3.61589645644142364110003434547, 4.56695105583377707290381398466, 5.80178002420682181857500886673, 6.25009314644919147894674491630, 6.74907060296788990548197331361, 7.85008231320428389902888482932, 8.40995512576165499767162638941, 9.22199198950462886837131324619, 9.69898729320740162673341290596, 11.13432280793717926216206148085, 11.45740002868701907825345946120, 12.27064341986663616685033057738, 12.97754153020133601602103714793, 13.409586904027088368431295225929, 14.33102330957966285196097104688, 14.94905665367909776290325304803, 15.74674245845383490443492479574, 16.47045246053025476770391134925, 17.340684635774064640801079116651, 17.96829213734615601776269625254, 18.53956665107981776069358113965, 19.14150953753721582082680197630

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