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Properties

Label 2-280-280.67-c1-0-17
Degree $2$
Conductor $280$
Sign $0.715 - 0.698i$
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  + (1.26 − 0.624i)2-s + (0.552 + 2.06i)3-s + (1.22 − 1.58i)4-s + (−1.28 + 1.82i)5-s + (1.98 + 2.27i)6-s + (0.126 + 2.64i)7-s + (0.559 − 2.77i)8-s + (−1.34 + 0.775i)9-s + (−0.491 + 3.12i)10-s + (−0.610 + 1.05i)11-s + (3.93 + 1.64i)12-s + (−2.68 − 2.68i)13-s + (1.81 + 3.27i)14-s + (−4.47 − 1.64i)15-s + (−1.02 − 3.86i)16-s + (6.90 − 1.84i)17-s + ⋯
L(s)  = 1  + (0.897 − 0.441i)2-s + (0.318 + 1.18i)3-s + (0.610 − 0.792i)4-s + (−0.575 + 0.817i)5-s + (0.811 + 0.926i)6-s + (0.0477 + 0.998i)7-s + (0.197 − 0.980i)8-s + (−0.447 + 0.258i)9-s + (−0.155 + 0.987i)10-s + (−0.183 + 0.318i)11-s + (1.13 + 0.473i)12-s + (−0.745 − 0.745i)13-s + (0.483 + 0.875i)14-s + (−1.15 − 0.424i)15-s + (−0.255 − 0.966i)16-s + (1.67 − 0.448i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.715 - 0.698i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.715 - 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.715 - 0.698i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.715 - 0.698i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.97648 + 0.804184i\)
\(L(\frac12)\) \(\approx\) \(1.97648 + 0.804184i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.26 + 0.624i)T \)
5 \( 1 + (1.28 - 1.82i)T \)
7 \( 1 + (-0.126 - 2.64i)T \)
good3 \( 1 + (-0.552 - 2.06i)T + (-2.59 + 1.5i)T^{2} \)
11 \( 1 + (0.610 - 1.05i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.68 + 2.68i)T + 13iT^{2} \)
17 \( 1 + (-6.90 + 1.84i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-0.485 + 0.280i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.57 + 5.88i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + 6.36T + 29T^{2} \)
31 \( 1 + (-5.87 - 3.38i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.32 - 1.15i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 11.6T + 41T^{2} \)
43 \( 1 + (-3.61 + 3.61i)T - 43iT^{2} \)
47 \( 1 + (2.05 + 0.551i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (3.77 - 1.01i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (2.73 + 1.58i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.85 + 2.22i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.46 + 0.392i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 3.28iT - 71T^{2} \)
73 \( 1 + (0.877 + 3.27i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-4.28 - 7.42i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (9.39 - 9.39i)T - 83iT^{2} \)
89 \( 1 + (12.2 - 7.08i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.07 + 7.07i)T + 97iT^{2} \)
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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

−12.05594783698422599244814002392, −11.02227123155560314270565053096, −10.13912661745856466976264626621, −9.641204745788059494462743043743, −8.141113173265748551973701769543, −6.90932876586329088553904050499, −5.51641048118213004878427794072, −4.69774238866898734946370131108, −3.39406604810601180906302713004, −2.70753882636816150858783381788, 1.49975892834312393701172185243, 3.36267489710609238192014514990, 4.51157836155491749151020448076, 5.71995676906160475198200488079, 7.06983461784297547887293472687, 7.64252102776289079672267271986, 8.245272497715992503080869566027, 9.788040617159987798015003076303, 11.36550506762893669052066963504, 12.03240426602571666945692041115

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