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Properties

Label 2-280-280.173-c1-0-12
Degree $2$
Conductor $280$
Sign $-0.347 - 0.937i$
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 + (0.133 − 0.5i)3-s + 2i·4-s + (−1.86 + 1.23i)5-s + (0.633 − 0.366i)6-s + (−0.866 + 2.5i)7-s + (−2 + 2i)8-s + (2.36 + 1.36i)9-s + (−3.09 − 0.633i)10-s + (2.36 − 1.36i)11-s + (1 + 0.267i)12-s + (−3 − 3i)13-s + (−3.36 + 1.63i)14-s + (0.366 + 1.09i)15-s − 4·16-s + (−1.09 + 4.09i)17-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + (0.0773 − 0.288i)3-s + i·4-s + (−0.834 + 0.550i)5-s + (0.258 − 0.149i)6-s + (−0.327 + 0.944i)7-s + (−0.707 + 0.707i)8-s + (0.788 + 0.455i)9-s + (−0.979 − 0.200i)10-s + (0.713 − 0.411i)11-s + (0.288 + 0.0773i)12-s + (−0.832 − 0.832i)13-s + (−0.899 + 0.436i)14-s + (0.0945 + 0.283i)15-s − 16-s + (−0.266 + 0.993i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.347 - 0.937i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.347 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.347 - 0.937i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ -0.347 - 0.937i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.910917 + 1.30938i\)
\(L(\frac12)\) \(\approx\) \(0.910917 + 1.30938i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.86 - 1.23i)T \)
7 \( 1 + (0.866 - 2.5i)T \)
good3 \( 1 + (-0.133 + 0.5i)T + (-2.59 - 1.5i)T^{2} \)
11 \( 1 + (-2.36 + 1.36i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (3 + 3i)T + 13iT^{2} \)
17 \( 1 + (1.09 - 4.09i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-3 - 1.73i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-5.59 + 1.5i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 2.46T + 29T^{2} \)
31 \( 1 + (0.169 - 0.0980i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.73 + 0.464i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 6.46iT - 41T^{2} \)
43 \( 1 + (-0.633 + 0.633i)T - 43iT^{2} \)
47 \( 1 + (-8.83 + 2.36i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (11.1 + 3i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-4.26 + 2.46i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.96 - 3.40i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.13 - 7.96i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 8.53T + 71T^{2} \)
73 \( 1 + (10.8 + 2.90i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-0.803 - 0.464i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.56 - 7.56i)T + 83iT^{2} \)
89 \( 1 + (-3.40 + 5.89i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (9.73 + 9.73i)T + 97iT^{2} \)
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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

−12.41284149295987391650113029350, −11.54518669762168911701888426422, −10.39419216567966950959691109409, −8.953287301961994851027625537752, −8.013393780779249849632467863231, −7.18891416598227618562113383834, −6.29252902301538921845250620458, −5.10573423713970040928633239350, −3.82254502998517373614979056898, −2.68270785133712729551857699180, 1.07820900346388312666851896048, 3.21494422213599849990499540359, 4.35721761208681542601523186705, 4.78478021936815025531800260754, 6.74540192067330739695182476977, 7.34015116366196970148201708593, 9.376209798241779414258866270483, 9.487760751155772441481019782998, 10.85354553709507605621061668946, 11.71208700042631971819077592899

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