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Properties

Label 2-280-40.29-c1-0-25
Degree $2$
Conductor $280$
Sign $0.268 + 0.963i$
Analytic cond. $2.23581$
Root an. cond. $1.49526$
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  + (−0.139 + 1.40i)2-s − 0.319·3-s + (−1.96 − 0.391i)4-s + (−1.20 − 1.88i)5-s + (0.0444 − 0.450i)6-s + i·7-s + (0.823 − 2.70i)8-s − 2.89·9-s + (2.82 − 1.42i)10-s − 4.31i·11-s + (0.627 + 0.125i)12-s − 2.16·13-s + (−1.40 − 0.139i)14-s + (0.384 + 0.603i)15-s + (3.69 + 1.53i)16-s − 3.19i·17-s + ⋯
L(s)  = 1  + (−0.0983 + 0.995i)2-s − 0.184·3-s + (−0.980 − 0.195i)4-s + (−0.537 − 0.843i)5-s + (0.0181 − 0.183i)6-s + 0.377i·7-s + (0.291 − 0.956i)8-s − 0.965·9-s + (0.892 − 0.451i)10-s − 1.30i·11-s + (0.181 + 0.0361i)12-s − 0.600·13-s + (−0.376 − 0.0371i)14-s + (0.0992 + 0.155i)15-s + (0.923 + 0.383i)16-s − 0.775i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.268 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.268 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.268 + 0.963i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.268 + 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.382329 - 0.290271i\)
\(L(\frac12)\) \(\approx\) \(0.382329 - 0.290271i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.139 - 1.40i)T \)
5 \( 1 + (1.20 + 1.88i)T \)
7 \( 1 - iT \)
good3 \( 1 + 0.319T + 3T^{2} \)
11 \( 1 + 4.31iT - 11T^{2} \)
13 \( 1 + 2.16T + 13T^{2} \)
17 \( 1 + 3.19iT - 17T^{2} \)
19 \( 1 + 5.38iT - 19T^{2} \)
23 \( 1 + 0.947iT - 23T^{2} \)
29 \( 1 - 7.55iT - 29T^{2} \)
31 \( 1 + 1.97T + 31T^{2} \)
37 \( 1 + 9.35T + 37T^{2} \)
41 \( 1 - 8.13T + 41T^{2} \)
43 \( 1 + 2.27T + 43T^{2} \)
47 \( 1 + 6.88iT - 47T^{2} \)
53 \( 1 + 6.12T + 53T^{2} \)
59 \( 1 + 4.30iT - 59T^{2} \)
61 \( 1 + 0.0705iT - 61T^{2} \)
67 \( 1 + 10.1T + 67T^{2} \)
71 \( 1 - 5.61T + 71T^{2} \)
73 \( 1 + 2.18iT - 73T^{2} \)
79 \( 1 - 10.0T + 79T^{2} \)
83 \( 1 - 16.4T + 83T^{2} \)
89 \( 1 + 13.4T + 89T^{2} \)
97 \( 1 + 11.4iT - 97T^{2} \)
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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

−11.73635241995780096751761436255, −10.80024873968696765746163081440, −9.141025925509426763603867648304, −8.842211556842824735680235771280, −7.86655211957521725246890646696, −6.72391289837924552280125520031, −5.47374088729804943437319480128, −4.92345733017247211826197033818, −3.29080099671076527121348586738, −0.37658754799680082125185374079, 2.15563052763292120311026190451, 3.48972361052177539803066834464, 4.53188810208794569636498048040, 5.97387780158261361473526667991, 7.42959903700445074609279517911, 8.195898753238910600445253055032, 9.572380284505712172795612663113, 10.34123542192666809412149488668, 11.07789795499762371227870443677, 12.03781691762211573781955438499

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