Properties

Label 2-280-35.17-c1-0-5
Degree $2$
Conductor $280$
Sign $0.994 - 0.108i$
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  + (1.66 − 0.447i)3-s + (2.21 + 0.298i)5-s + (0.900 + 2.48i)7-s + (−0.0138 + 0.00800i)9-s + (−1.10 + 1.91i)11-s + (−1.91 − 1.91i)13-s + (3.83 − 0.491i)15-s + (−1.12 − 4.19i)17-s + (−1.80 − 3.12i)19-s + (2.61 + 3.74i)21-s + (0.375 + 0.100i)23-s + (4.82 + 1.32i)25-s + (−3.68 + 3.68i)27-s − 6.62i·29-s + (0.897 + 0.518i)31-s + ⋯
L(s)  = 1  + (0.963 − 0.258i)3-s + (0.991 + 0.133i)5-s + (0.340 + 0.940i)7-s + (−0.00462 + 0.00266i)9-s + (−0.333 + 0.578i)11-s + (−0.530 − 0.530i)13-s + (0.989 − 0.127i)15-s + (−0.272 − 1.01i)17-s + (−0.413 − 0.716i)19-s + (0.570 + 0.817i)21-s + (0.0783 + 0.0209i)23-s + (0.964 + 0.264i)25-s + (−0.708 + 0.708i)27-s − 1.22i·29-s + (0.161 + 0.0930i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.108i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.108i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.994 - 0.108i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.994 - 0.108i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.87783 + 0.101703i\)
\(L(\frac12)\) \(\approx\) \(1.87783 + 0.101703i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.21 - 0.298i)T \)
7 \( 1 + (-0.900 - 2.48i)T \)
good3 \( 1 + (-1.66 + 0.447i)T + (2.59 - 1.5i)T^{2} \)
11 \( 1 + (1.10 - 1.91i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.91 + 1.91i)T + 13iT^{2} \)
17 \( 1 + (1.12 + 4.19i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (1.80 + 3.12i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.375 - 0.100i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + 6.62iT - 29T^{2} \)
31 \( 1 + (-0.897 - 0.518i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.84 + 6.88i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 4.03iT - 41T^{2} \)
43 \( 1 + (8.37 - 8.37i)T - 43iT^{2} \)
47 \( 1 + (-5.52 - 1.48i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-1.62 - 6.07i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-3.71 + 6.42i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.84 + 4.52i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (10.0 - 2.70i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 9.39T + 71T^{2} \)
73 \( 1 + (13.7 - 3.69i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-5.38 + 3.11i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.77 - 7.77i)T + 83iT^{2} \)
89 \( 1 + (-5.20 - 9.01i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (12.1 - 12.1i)T - 97iT^{2} \)
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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

−11.98688242485529197172928701290, −10.87875369585194682344880682976, −9.679170060317167321458416326032, −9.095154615055607764019390057697, −8.118213851526890409894559831266, −7.12088696440841854524980363840, −5.78838524164624799487554021059, −4.83606377335671619185981552334, −2.75492855930375546857233662021, −2.21591204776973086227417970130, 1.81125366035995522035335737212, 3.24836573776926843298366189306, 4.47912983954489465897038621581, 5.82785935844782355012351436663, 6.98613542246045609662838373934, 8.280558693854108670159107220844, 8.859498322762244304542250395269, 10.03972804343650294452008277998, 10.53720697084875460384763363954, 11.83482341641431181223491495753

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