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Properties

Label 2-280-40.27-c1-0-3
Degree $2$
Conductor $280$
Sign $-0.276 - 0.960i$
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  + (−1.15 − 0.814i)2-s + (−0.824 + 0.824i)3-s + (0.672 + 1.88i)4-s + (1.57 + 1.58i)5-s + (1.62 − 0.281i)6-s + (−0.707 + 0.707i)7-s + (0.757 − 2.72i)8-s + 1.63i·9-s + (−0.533 − 3.11i)10-s − 3.75·11-s + (−2.10 − 0.999i)12-s + (−4.02 − 4.02i)13-s + (1.39 − 0.241i)14-s + (−2.60 − 0.00463i)15-s + (−3.09 + 2.53i)16-s + (3.23 + 3.23i)17-s + ⋯
L(s)  = 1  + (−0.817 − 0.576i)2-s + (−0.476 + 0.476i)3-s + (0.336 + 0.941i)4-s + (0.705 + 0.708i)5-s + (0.663 − 0.114i)6-s + (−0.267 + 0.267i)7-s + (0.267 − 0.963i)8-s + 0.546i·9-s + (−0.168 − 0.985i)10-s − 1.13·11-s + (−0.608 − 0.288i)12-s + (−1.11 − 1.11i)13-s + (0.372 − 0.0644i)14-s + (−0.673 − 0.00119i)15-s + (−0.774 + 0.633i)16-s + (0.784 + 0.784i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.276 - 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.276 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.276 - 0.960i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (267, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ -0.276 - 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.338651 + 0.450035i\)
\(L(\frac12)\) \(\approx\) \(0.338651 + 0.450035i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.15 + 0.814i)T \)
5 \( 1 + (-1.57 - 1.58i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
good3 \( 1 + (0.824 - 0.824i)T - 3iT^{2} \)
11 \( 1 + 3.75T + 11T^{2} \)
13 \( 1 + (4.02 + 4.02i)T + 13iT^{2} \)
17 \( 1 + (-3.23 - 3.23i)T + 17iT^{2} \)
19 \( 1 - 7.11iT - 19T^{2} \)
23 \( 1 + (2.28 + 2.28i)T + 23iT^{2} \)
29 \( 1 - 3.96T + 29T^{2} \)
31 \( 1 - 1.00iT - 31T^{2} \)
37 \( 1 + (7.48 - 7.48i)T - 37iT^{2} \)
41 \( 1 - 4.39T + 41T^{2} \)
43 \( 1 + (4.93 - 4.93i)T - 43iT^{2} \)
47 \( 1 + (-4.78 + 4.78i)T - 47iT^{2} \)
53 \( 1 + (-6.78 - 6.78i)T + 53iT^{2} \)
59 \( 1 - 0.808iT - 59T^{2} \)
61 \( 1 + 12.9iT - 61T^{2} \)
67 \( 1 + (1.01 + 1.01i)T + 67iT^{2} \)
71 \( 1 - 4.28iT - 71T^{2} \)
73 \( 1 + (-2.38 + 2.38i)T - 73iT^{2} \)
79 \( 1 - 11.6T + 79T^{2} \)
83 \( 1 + (-5.74 + 5.74i)T - 83iT^{2} \)
89 \( 1 - 4.29iT - 89T^{2} \)
97 \( 1 + (-5.24 - 5.24i)T + 97iT^{2} \)
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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

−12.06550572432275946019863568901, −10.67085916885367807945997517301, −10.22795202738361048153710026639, −9.929868452288364427415852848976, −8.242941271359843619145661461970, −7.60223957727025320751538413513, −6.12467781297711069288459151011, −5.13209358520541468795730339665, −3.30246766654960390152744151395, −2.19847366352240328492149027919, 0.55649646235253790781950300840, 2.29941109078755098505971387387, 4.86480749274964308182235203038, 5.66585318142823907994638694569, 6.85322518072727428489353634727, 7.45075912580846921888683926094, 8.890224251274812243142618822509, 9.531910661184685806355765406135, 10.35008559052887210705146197306, 11.60147506392201027520506199207

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