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Properties

Label 2-280-280.213-c1-0-6
Degree $2$
Conductor $280$
Sign $-0.979 + 0.199i$
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  + (0.667 + 1.24i)2-s + (−2.38 + 0.637i)3-s + (−1.10 + 1.66i)4-s + (2.06 + 0.861i)5-s + (−2.38 − 2.54i)6-s + (−1.15 + 2.37i)7-s + (−2.81 − 0.268i)8-s + (2.66 − 1.53i)9-s + (0.304 + 3.14i)10-s + (1.20 + 0.694i)11-s + (1.57 − 4.66i)12-s + (−4.02 − 4.02i)13-s + (−3.73 + 0.146i)14-s + (−5.46 − 0.733i)15-s + (−1.54 − 3.68i)16-s + (−1.42 + 0.382i)17-s + ⋯
L(s)  = 1  + (0.472 + 0.881i)2-s + (−1.37 + 0.368i)3-s + (−0.553 + 0.832i)4-s + (0.922 + 0.385i)5-s + (−0.973 − 1.03i)6-s + (−0.437 + 0.899i)7-s + (−0.995 − 0.0949i)8-s + (0.886 − 0.511i)9-s + (0.0964 + 0.995i)10-s + (0.362 + 0.209i)11-s + (0.454 − 1.34i)12-s + (−1.11 − 1.11i)13-s + (−0.999 + 0.0392i)14-s + (−1.40 − 0.189i)15-s + (−0.386 − 0.922i)16-s + (−0.346 + 0.0927i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 + 0.199i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.979 + 0.199i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.979 + 0.199i$
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: \((2,\ 280,\ (\ :1/2),\ -0.979 + 0.199i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0804031 - 0.797693i\)
\(L(\frac12)\) \(\approx\) \(0.0804031 - 0.797693i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.667 - 1.24i)T \)
5 \( 1 + (-2.06 - 0.861i)T \)
7 \( 1 + (1.15 - 2.37i)T \)
good3 \( 1 + (2.38 - 0.637i)T + (2.59 - 1.5i)T^{2} \)
11 \( 1 + (-1.20 - 0.694i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.02 + 4.02i)T + 13iT^{2} \)
17 \( 1 + (1.42 - 0.382i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (3.07 - 1.77i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.32 - 4.94i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 - 7.04T + 29T^{2} \)
31 \( 1 + (-2.00 - 1.15i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.11 - 7.89i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 5.68iT - 41T^{2} \)
43 \( 1 + (-1.18 + 1.18i)T - 43iT^{2} \)
47 \( 1 + (-1.27 + 4.75i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-3.37 - 12.6i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (4.56 + 2.63i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.71 - 9.89i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.01 + 1.61i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 1.25T + 71T^{2} \)
73 \( 1 + (-2.00 - 7.49i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (1.75 - 1.01i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (8.66 + 8.66i)T + 83iT^{2} \)
89 \( 1 + (5.66 + 9.80i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.47 - 1.47i)T + 97iT^{2} \)
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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

−12.34934006613424817899395618316, −11.71301986031312105279287455493, −10.31164011644659982320449250549, −9.721735495558513133391061824365, −8.471551865152943621087032324091, −6.99139028950050183086225290948, −6.13551452782940998253613865990, −5.55552447728952953476731662023, −4.67597046366746206423200534368, −2.85695915238612263636485963756, 0.60339799433082511673185273639, 2.21191611428472269526436770848, 4.29005107373133487980894318252, 5.06572005434351232355119693461, 6.32847459198475629052537029531, 6.77545483789632459414980954077, 8.889125521601256410489455312440, 9.877522294385850730102597777292, 10.58468251796216269738605223974, 11.41620565659297442182167590459

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