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Properties

Label 2-280-5.4-c5-0-30
Degree $2$
Conductor $280$
Sign $0.911 + 0.411i$
Analytic cond. $44.9074$
Root an. cond. $6.70130$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.606i·3-s + (50.9 + 23.0i)5-s + 49i·7-s + 242.·9-s − 116.·11-s − 1.12e3i·13-s + (13.9 − 30.8i)15-s − 1.27e3i·17-s + 981.·19-s + 29.7·21-s + 1.94e3i·23-s + (2.06e3 + 2.34e3i)25-s − 294. i·27-s − 7.65e3·29-s + 8.73e3·31-s + ⋯
L(s)  = 1  − 0.0388i·3-s + (0.911 + 0.411i)5-s + 0.377i·7-s + 0.998·9-s − 0.291·11-s − 1.84i·13-s + (0.0160 − 0.0354i)15-s − 1.07i·17-s + 0.623·19-s + 0.0146·21-s + 0.765i·23-s + (0.660 + 0.750i)25-s − 0.0777i·27-s − 1.68·29-s + 1.63·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 + 0.411i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.911 + 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.911 + 0.411i$
Analytic conductor: \(44.9074\)
Root analytic conductor: \(6.70130\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :5/2),\ 0.911 + 0.411i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.658852494\)
\(L(\frac12)\) \(\approx\) \(2.658852494\)
\(L(\frac{7}{2})\) 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 \( 1 + (-50.9 - 23.0i)T \)
7 \( 1 - 49iT \)
good3 \( 1 + 0.606iT - 243T^{2} \)
11 \( 1 + 116.T + 1.61e5T^{2} \)
13 \( 1 + 1.12e3iT - 3.71e5T^{2} \)
17 \( 1 + 1.27e3iT - 1.41e6T^{2} \)
19 \( 1 - 981.T + 2.47e6T^{2} \)
23 \( 1 - 1.94e3iT - 6.43e6T^{2} \)
29 \( 1 + 7.65e3T + 2.05e7T^{2} \)
31 \( 1 - 8.73e3T + 2.86e7T^{2} \)
37 \( 1 + 8.23e3iT - 6.93e7T^{2} \)
41 \( 1 - 3.97e3T + 1.15e8T^{2} \)
43 \( 1 - 3.29e3iT - 1.47e8T^{2} \)
47 \( 1 + 1.35e4iT - 2.29e8T^{2} \)
53 \( 1 - 3.20e3iT - 4.18e8T^{2} \)
59 \( 1 - 3.58e4T + 7.14e8T^{2} \)
61 \( 1 - 1.46e3T + 8.44e8T^{2} \)
67 \( 1 - 9.42e3iT - 1.35e9T^{2} \)
71 \( 1 - 3.10e4T + 1.80e9T^{2} \)
73 \( 1 - 2.89e3iT - 2.07e9T^{2} \)
79 \( 1 - 6.08e4T + 3.07e9T^{2} \)
83 \( 1 + 1.15e5iT - 3.93e9T^{2} \)
89 \( 1 - 2.72e4T + 5.58e9T^{2} \)
97 \( 1 - 4.26e4iT - 8.58e9T^{2} \)
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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

−10.75934992403198984407694071811, −9.966199458469800490604545099033, −9.323861438084890028205262418358, −7.87471482598074899666145713003, −7.09739803143260340437913985130, −5.78845285223263099977617923222, −5.10827374221326364499138725963, −3.37015520397601891475142515391, −2.28359003518004476015659768115, −0.835823819675397543484547790025, 1.18775107060693882976768054397, 2.13709779555190148414196857149, 3.98650539933843203378026273452, 4.84704862277901646628666433008, 6.21208977121710314806485710028, 6.99358243651653127659298094290, 8.268052179625003506675472215199, 9.388341310166519179495231916446, 9.963559336031658642917917222076, 10.95712690971045992809894676980

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