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Properties

Label 2-280-280.117-c1-0-1
Degree $2$
Conductor $280$
Sign $-0.337 - 0.941i$
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  + (−0.751 + 1.19i)2-s + (−3.03 − 0.811i)3-s + (−0.870 − 1.80i)4-s + (−0.203 − 2.22i)5-s + (3.24 − 3.01i)6-s + (−2.64 − 0.167i)7-s + (2.81 + 0.310i)8-s + (5.92 + 3.42i)9-s + (2.82 + 1.42i)10-s + (−1.83 + 1.06i)11-s + (1.17 + 6.16i)12-s + (−0.498 + 0.498i)13-s + (2.18 − 3.03i)14-s + (−1.19 + 6.91i)15-s + (−2.48 + 3.13i)16-s + (−0.344 − 0.0923i)17-s + ⋯
L(s)  = 1  + (−0.531 + 0.847i)2-s + (−1.74 − 0.468i)3-s + (−0.435 − 0.900i)4-s + (−0.0909 − 0.995i)5-s + (1.32 − 1.23i)6-s + (−0.997 − 0.0633i)7-s + (0.993 + 0.109i)8-s + (1.97 + 1.14i)9-s + (0.891 + 0.452i)10-s + (−0.553 + 0.319i)11-s + (0.339 + 1.77i)12-s + (−0.138 + 0.138i)13-s + (0.583 − 0.811i)14-s + (−0.307 + 1.78i)15-s + (−0.621 + 0.783i)16-s + (−0.0836 − 0.0224i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.337 - 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.337 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.337 - 0.941i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ -0.337 - 0.941i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.132562 + 0.188432i\)
\(L(\frac12)\) \(\approx\) \(0.132562 + 0.188432i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.751 - 1.19i)T \)
5 \( 1 + (0.203 + 2.22i)T \)
7 \( 1 + (2.64 + 0.167i)T \)
good3 \( 1 + (3.03 + 0.811i)T + (2.59 + 1.5i)T^{2} \)
11 \( 1 + (1.83 - 1.06i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.498 - 0.498i)T - 13iT^{2} \)
17 \( 1 + (0.344 + 0.0923i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-3.42 - 1.97i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.24 - 4.65i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 3.32T + 29T^{2} \)
31 \( 1 + (-5.09 + 2.94i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.23 - 8.34i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 6.13iT - 41T^{2} \)
43 \( 1 + (-6.70 - 6.70i)T + 43iT^{2} \)
47 \( 1 + (2.51 + 9.36i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.228 + 0.851i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (12.9 - 7.48i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.445 + 0.772i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-9.66 - 2.58i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 1.28T + 71T^{2} \)
73 \( 1 + (-0.278 + 1.04i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (11.4 + 6.61i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.33 - 5.33i)T - 83iT^{2} \)
89 \( 1 + (5.51 - 9.54i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.53 - 5.53i)T - 97iT^{2} \)
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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

−12.11550099410717801029636934572, −11.25932416686530712894755055351, −10.02693916924168960763080654442, −9.538084323058212792285706558805, −7.985020022946183383722799222869, −7.13714351414648620526348364199, −6.12611458274939049940392423783, −5.43093739529514586835155058974, −4.50209633209556344209030364093, −1.17357374366506997400987059606, 0.30363756201374798367106274770, 2.87965751303176690837414175908, 4.13730094812159853617939146938, 5.50911492670013322283487591350, 6.59364602990963856546049882627, 7.47426485509050834930890895135, 9.236235895712195394104997001452, 10.09542544211955690822483725860, 10.73365222289255879241224673335, 11.25516219250605710630445318674

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