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Properties

Label 2-280-40.29-c1-0-22
Degree $2$
Conductor $280$
Sign $-0.555 + 0.831i$
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  + (−1.36 − 0.363i)2-s + 0.460·3-s + (1.73 + 0.993i)4-s + (−2.19 + 0.423i)5-s + (−0.629 − 0.167i)6-s − i·7-s + (−2.01 − 1.98i)8-s − 2.78·9-s + (3.15 + 0.219i)10-s − 4.70i·11-s + (0.799 + 0.457i)12-s + 0.906·13-s + (−0.363 + 1.36i)14-s + (−1.01 + 0.195i)15-s + (2.02 + 3.44i)16-s − 7.25i·17-s + ⋯
L(s)  = 1  + (−0.966 − 0.257i)2-s + 0.266·3-s + (0.867 + 0.496i)4-s + (−0.981 + 0.189i)5-s + (−0.257 − 0.0683i)6-s − 0.377i·7-s + (−0.710 − 0.703i)8-s − 0.929·9-s + (0.997 + 0.0693i)10-s − 1.41i·11-s + (0.230 + 0.132i)12-s + 0.251·13-s + (−0.0971 + 0.365i)14-s + (−0.261 + 0.0504i)15-s + (0.506 + 0.862i)16-s − 1.75i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.555 + 0.831i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.555 + 0.831i$
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: \((2,\ 280,\ (\ :1/2),\ -0.555 + 0.831i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.226249 - 0.423429i\)
\(L(\frac12)\) \(\approx\) \(0.226249 - 0.423429i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 + 0.363i)T \)
5 \( 1 + (2.19 - 0.423i)T \)
7 \( 1 + iT \)
good3 \( 1 - 0.460T + 3T^{2} \)
11 \( 1 + 4.70iT - 11T^{2} \)
13 \( 1 - 0.906T + 13T^{2} \)
17 \( 1 + 7.25iT - 17T^{2} \)
19 \( 1 + 2.15iT - 19T^{2} \)
23 \( 1 - 5.89iT - 23T^{2} \)
29 \( 1 + 6.91iT - 29T^{2} \)
31 \( 1 + 8.02T + 31T^{2} \)
37 \( 1 + 6.21T + 37T^{2} \)
41 \( 1 + 6.21T + 41T^{2} \)
43 \( 1 - 9.32T + 43T^{2} \)
47 \( 1 - 3.80iT - 47T^{2} \)
53 \( 1 - 4.96T + 53T^{2} \)
59 \( 1 - 11.7iT - 59T^{2} \)
61 \( 1 + 6.20iT - 61T^{2} \)
67 \( 1 - 5.42T + 67T^{2} \)
71 \( 1 - 1.74T + 71T^{2} \)
73 \( 1 - 5.82iT - 73T^{2} \)
79 \( 1 - 8.05T + 79T^{2} \)
83 \( 1 + 10.5T + 83T^{2} \)
89 \( 1 - 11.5T + 89T^{2} \)
97 \( 1 + 11.6iT - 97T^{2} \)
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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

−11.36159894476588694167230777111, −10.85239335965478990543507741712, −9.416003561231916444378372878098, −8.689996398499746021220656359533, −7.80890088798885246078667709855, −7.02957111419379156173947843791, −5.63165874039831360372321594085, −3.67919259856188072739741970834, −2.83880038241106375756362394039, −0.46176611064920139313111161527, 1.98788357168117373212525140668, 3.63654446645892462501307954281, 5.25267207452554365247188797512, 6.53585294518321966332889763740, 7.57432963015893945201436519969, 8.480552832416314056917867692422, 8.970608478200695610415005869364, 10.33047794351158353852530029906, 11.01771168965377631866252962580, 12.18548421731232086584352155095

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