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Properties

Label 2-280-35.12-c1-0-3
Degree $2$
Conductor $280$
Sign $0.761 - 0.648i$
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  + (−0.0200 + 0.0749i)3-s + (−2.22 + 0.172i)5-s + (2.20 + 1.45i)7-s + (2.59 + 1.49i)9-s + (−0.390 − 0.677i)11-s + (3.28 + 3.28i)13-s + (0.0318 − 0.170i)15-s + (5.24 + 1.40i)17-s + (−2.91 + 5.05i)19-s + (−0.153 + 0.136i)21-s + (−2.24 − 8.39i)23-s + (4.94 − 0.770i)25-s + (−0.328 + 0.328i)27-s − 0.303i·29-s + (−5.04 + 2.91i)31-s + ⋯
L(s)  = 1  + (−0.0115 + 0.0432i)3-s + (−0.997 + 0.0772i)5-s + (0.834 + 0.551i)7-s + (0.864 + 0.498i)9-s + (−0.117 − 0.204i)11-s + (0.911 + 0.911i)13-s + (0.00821 − 0.0440i)15-s + (1.27 + 0.341i)17-s + (−0.669 + 1.15i)19-s + (−0.0335 + 0.0297i)21-s + (−0.468 − 1.74i)23-s + (0.988 − 0.154i)25-s + (−0.0632 + 0.0632i)27-s − 0.0564i·29-s + (−0.905 + 0.522i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.761 - 0.648i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.761 - 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.761 - 0.648i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.761 - 0.648i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16927 + 0.430486i\)
\(L(\frac12)\) \(\approx\) \(1.16927 + 0.430486i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.22 - 0.172i)T \)
7 \( 1 + (-2.20 - 1.45i)T \)
good3 \( 1 + (0.0200 - 0.0749i)T + (-2.59 - 1.5i)T^{2} \)
11 \( 1 + (0.390 + 0.677i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.28 - 3.28i)T + 13iT^{2} \)
17 \( 1 + (-5.24 - 1.40i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (2.91 - 5.05i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.24 + 8.39i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 0.303iT - 29T^{2} \)
31 \( 1 + (5.04 - 2.91i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (7.12 - 1.90i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 4.39iT - 41T^{2} \)
43 \( 1 + (-7.33 + 7.33i)T - 43iT^{2} \)
47 \( 1 + (-0.611 - 2.28i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-6.01 - 1.61i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (2.56 + 4.43i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (8.91 + 5.14i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.395 + 1.47i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 5.83T + 71T^{2} \)
73 \( 1 + (-3.47 + 12.9i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (8.66 + 5.00i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.94 + 2.94i)T + 83iT^{2} \)
89 \( 1 + (6.43 - 11.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.0900 + 0.0900i)T - 97iT^{2} \)
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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

−12.20073438611759307231950355876, −10.89660566417407707078236023780, −10.42456588741235054462065844867, −8.806732487322628908436377345640, −8.176700144261140781432996102598, −7.25254528658902473677409325155, −5.94479391189714180251640023090, −4.61408721277936421815283263326, −3.71102090472531164118631179429, −1.77115865036193023644332868976, 1.14723723277361709548473216527, 3.43801456320211690863188744303, 4.35902251050868485913311189767, 5.60084027419740522856118300916, 7.23291807012060123645599071745, 7.67069720037328080181110479459, 8.754991407039310654867716287958, 9.981189171390589039939129556437, 10.95402549682191845694684827833, 11.65433664164162374219517615590

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