Properties

Label 2-280-280.213-c1-0-25
Degree $2$
Conductor $280$
Sign $0.730 - 0.683i$
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 + i)2-s + (1.86 − 0.5i)3-s + 2i·4-s + (−0.133 − 2.23i)5-s + (2.36 + 1.36i)6-s + (0.866 + 2.5i)7-s + (−2 + 2i)8-s + (0.633 − 0.366i)9-s + (2.09 − 2.36i)10-s + (0.633 + 0.366i)11-s + (1 + 3.73i)12-s + (−3 − 3i)13-s + (−1.63 + 3.36i)14-s + (−1.36 − 4.09i)15-s − 4·16-s + (4.09 − 1.09i)17-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + (1.07 − 0.288i)3-s + i·4-s + (−0.0599 − 0.998i)5-s + (0.965 + 0.557i)6-s + (0.327 + 0.944i)7-s + (−0.707 + 0.707i)8-s + (0.211 − 0.122i)9-s + (0.663 − 0.748i)10-s + (0.191 + 0.110i)11-s + (0.288 + 1.07i)12-s + (−0.832 − 0.832i)13-s + (−0.436 + 0.899i)14-s + (−0.352 − 1.05i)15-s − 16-s + (0.993 − 0.266i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.730 - 0.683i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.730 - 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.730 - 0.683i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (213, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.730 - 0.683i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.20886 + 0.872017i\)
\(L(\frac12)\) \(\approx\) \(2.20886 + 0.872017i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - i)T \)
5 \( 1 + (0.133 + 2.23i)T \)
7 \( 1 + (-0.866 - 2.5i)T \)
good3 \( 1 + (-1.86 + 0.5i)T + (2.59 - 1.5i)T^{2} \)
11 \( 1 + (-0.633 - 0.366i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (3 + 3i)T + 13iT^{2} \)
17 \( 1 + (-4.09 + 1.09i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-3 + 1.73i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.401 + 1.5i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + 4.46T + 29T^{2} \)
31 \( 1 + (8.83 + 5.09i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.73 - 6.46i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 0.464iT - 41T^{2} \)
43 \( 1 + (-2.36 + 2.36i)T - 43iT^{2} \)
47 \( 1 + (-0.169 + 0.633i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (0.803 + 3i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-7.73 - 4.46i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.96 - 8.59i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.86 - 1.03i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 15.4T + 71T^{2} \)
73 \( 1 + (2.16 + 8.09i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-11.1 + 6.46i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.56 + 4.56i)T + 83iT^{2} \)
89 \( 1 + (-8.59 - 14.8i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.26 + 6.26i)T + 97iT^{2} \)
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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

−12.31252538635482516093107056128, −11.52553834261788530549363034143, −9.547962308936544748566146008221, −8.856150741884632480976051535263, −7.985806684140267551315960356137, −7.41270832456188885269580035708, −5.63524496723586120461980288714, −5.06651400597055332186517501619, −3.51373938271878839443148989183, −2.30885854148395111787779706013, 1.98976639421233728469804954924, 3.38019176946268871953115662208, 3.90394222983684546146223926867, 5.44865415767917462709124295649, 6.91478729484676280178054894713, 7.74755342338894154378820966290, 9.291416231105555313481487378202, 9.913760007070162298129485408246, 10.87186275066179878525744190781, 11.62988828876085233461159122883

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