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Properties

Label 2-280-35.27-c1-0-11
Degree $2$
Conductor $280$
Sign $-0.894 + 0.447i$
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  + (−2.16 − 2.16i)3-s + (1.91 − 1.14i)5-s + (−0.409 − 2.61i)7-s + 6.36i·9-s + 0.796·11-s + (−3.25 − 3.25i)13-s + (−6.63 − 1.66i)15-s + (−2.52 + 2.52i)17-s − 2.29·19-s + (−4.76 + 6.54i)21-s + (−2.08 + 2.08i)23-s + (2.35 − 4.40i)25-s + (7.27 − 7.27i)27-s − 10.0i·29-s + 3.63i·31-s + ⋯
L(s)  = 1  + (−1.24 − 1.24i)3-s + (0.857 − 0.513i)5-s + (−0.154 − 0.987i)7-s + 2.12i·9-s + 0.240·11-s + (−0.901 − 0.901i)13-s + (−1.71 − 0.429i)15-s + (−0.611 + 0.611i)17-s − 0.527·19-s + (−1.04 + 1.42i)21-s + (−0.434 + 0.434i)23-s + (0.471 − 0.881i)25-s + (1.39 − 1.39i)27-s − 1.87i·29-s + 0.652i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.175623 - 0.744320i\)
\(L(\frac12)\) \(\approx\) \(0.175623 - 0.744320i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-1.91 + 1.14i)T \)
7 \( 1 + (0.409 + 2.61i)T \)
good3 \( 1 + (2.16 + 2.16i)T + 3iT^{2} \)
11 \( 1 - 0.796T + 11T^{2} \)
13 \( 1 + (3.25 + 3.25i)T + 13iT^{2} \)
17 \( 1 + (2.52 - 2.52i)T - 17iT^{2} \)
19 \( 1 + 2.29T + 19T^{2} \)
23 \( 1 + (2.08 - 2.08i)T - 23iT^{2} \)
29 \( 1 + 10.0iT - 29T^{2} \)
31 \( 1 - 3.63iT - 31T^{2} \)
37 \( 1 + (-7.30 - 7.30i)T + 37iT^{2} \)
41 \( 1 + 2.81iT - 41T^{2} \)
43 \( 1 + (-2.33 + 2.33i)T - 43iT^{2} \)
47 \( 1 + (-4.09 + 4.09i)T - 47iT^{2} \)
53 \( 1 + (-6.50 + 6.50i)T - 53iT^{2} \)
59 \( 1 - 11.4T + 59T^{2} \)
61 \( 1 - 3.35iT - 61T^{2} \)
67 \( 1 + (2.49 + 2.49i)T + 67iT^{2} \)
71 \( 1 + 2.93T + 71T^{2} \)
73 \( 1 + (4.93 + 4.93i)T + 73iT^{2} \)
79 \( 1 + 6.53iT - 79T^{2} \)
83 \( 1 + (1.14 + 1.14i)T + 83iT^{2} \)
89 \( 1 - 14.0T + 89T^{2} \)
97 \( 1 + (4.30 - 4.30i)T - 97iT^{2} \)
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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

−11.64213021390088894701478278669, −10.53404448226542500397839648973, −9.913921631117352204001669098528, −8.295647964532179137929426360046, −7.32456061595399255126555172529, −6.40111243909175969146702555806, −5.65000957904368876988171946304, −4.49061794564114235235281090066, −2.10498334830865071284764785332, −0.66154995997107720585091666123, 2.51177427769847085645836978679, 4.25313866634779947939681877485, 5.26028573569612735957718322603, 6.08319733853609652360710629554, 6.92438110373472062308977200497, 9.080179444513395679357610181576, 9.456413829315671885751985403771, 10.43451014330730965347920625225, 11.22186207923883466425403899363, 11.97566892751767088516489800684

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