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Properties

Label 2-280-40.29-c1-0-8
Degree $2$
Conductor $280$
Sign $-0.759 - 0.650i$
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 + (−2.48 + 1.95i)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 + (−0.786 + 0.617i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.759 - 0.650i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.759 - 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.759 - 0.650i$
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: \((2,\ 280,\ (\ :1/2),\ -0.759 - 0.650i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.434352 + 1.17533i\)
\(L(\frac12)\) \(\approx\) \(0.434352 + 1.17533i\)
\(L(\frac{3}{2})\) 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

−12.38076507365268564277254544082, −11.31339222642224326562614109987, −9.979175012600355149768379313630, −9.370418709207096016081847493336, −8.199482558455954097330274791903, −7.31246058971033329272447788465, −6.21424288261922467758629255716, −5.51648257102869936327641215793, −4.01980981582528602356318241885, −2.52511927699076133680665729823, 0.972859303285141930431100021733, 2.71499916710746273537119939890, 3.94478451381160323748289352826, 5.28251860640526799924424884198, 6.11342768080119087172888900151, 8.150423684554249152395827271633, 8.811974337347438150919110196073, 9.502365741075600056389924938441, 10.95622131521580435503196585805, 11.17106265461579610792848453703

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