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Properties

Label 2-280-7.4-c3-0-7
Degree $2$
Conductor $280$
Sign $-0.991 + 0.126i$
Analytic cond. $16.5205$
Root an. cond. $4.06454$
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  + (−3.5 + 6.06i)3-s + (2.5 + 4.33i)5-s + (3.5 + 18.1i)7-s + (−11 − 19.0i)9-s + (−29 + 50.2i)11-s + 82·13-s − 35·15-s + (−25 + 43.3i)17-s + (−32 − 55.4i)19-s + (−122.5 − 42.4i)21-s + (55.5 + 96.1i)23-s + (−12.5 + 21.6i)25-s − 35.0·27-s + 103·29-s + (65 − 112. i)31-s + ⋯
L(s)  = 1  + (−0.673 + 1.16i)3-s + (0.223 + 0.387i)5-s + (0.188 + 0.981i)7-s + (−0.407 − 0.705i)9-s + (−0.794 + 1.37i)11-s + 1.74·13-s − 0.602·15-s + (−0.356 + 0.617i)17-s + (−0.386 − 0.669i)19-s + (−1.27 − 0.440i)21-s + (0.503 + 0.871i)23-s + (−0.100 + 0.173i)25-s − 0.249·27-s + 0.659·29-s + (0.376 − 0.652i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.991 + 0.126i$
Analytic conductor: \(16.5205\)
Root analytic conductor: \(4.06454\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :3/2),\ -0.991 + 0.126i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.182461565\)
\(L(\frac12)\) \(\approx\) \(1.182461565\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.5 - 4.33i)T \)
7 \( 1 + (-3.5 - 18.1i)T \)
good3 \( 1 + (3.5 - 6.06i)T + (-13.5 - 23.3i)T^{2} \)
11 \( 1 + (29 - 50.2i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 82T + 2.19e3T^{2} \)
17 \( 1 + (25 - 43.3i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (32 + 55.4i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-55.5 - 96.1i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 103T + 2.43e4T^{2} \)
31 \( 1 + (-65 + 112. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (188 + 325. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 307T + 6.89e4T^{2} \)
43 \( 1 + 197T + 7.95e4T^{2} \)
47 \( 1 + (60 + 103. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-254 + 439. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (300 - 519. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-82.5 - 142. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-316.5 + 548. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 840T + 3.57e5T^{2} \)
73 \( 1 + (303 - 524. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-658 - 1.13e3i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 61T + 5.71e5T^{2} \)
89 \( 1 + (-93.5 - 161. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 406T + 9.12e5T^{2} \)
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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

−11.59560752761809697768659222039, −10.89034930035001721152852817513, −10.18100299253983700917528019161, −9.248393436704561328086756911518, −8.284062543157981878091260833824, −6.76478147636842431553882527762, −5.67745049379818177983647645963, −4.89795716314073745381313288440, −3.69857540536067335720712030590, −2.08338995306755589223515431297, 0.52185977981391718091680818150, 1.41318428586734288907352820178, 3.36276871556921022259308770403, 4.95362632355586402340829423380, 6.15926143775471184016708559026, 6.73044293522881481754444403738, 8.080120498539775468298745083400, 8.612740752605213903175287420079, 10.36271321260575420694687390688, 10.99964615122909534875762087810

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