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Properties

Label 2-280-35.13-c1-0-9
Degree $2$
Conductor $280$
Sign $0.690 + 0.723i$
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  + (1.41 − 1.41i)3-s + (1.94 − 1.09i)5-s + (2.64 − 0.0496i)7-s − 0.987i·9-s − 5.75·11-s + (−2.89 + 2.89i)13-s + (1.20 − 4.29i)15-s + (2.13 + 2.13i)17-s + 2.18·19-s + (3.66 − 3.80i)21-s + (−4.79 − 4.79i)23-s + (2.60 − 4.26i)25-s + (2.84 + 2.84i)27-s + 5.19i·29-s − 6.68i·31-s + ⋯
L(s)  = 1  + (0.815 − 0.815i)3-s + (0.871 − 0.489i)5-s + (0.999 − 0.0187i)7-s − 0.329i·9-s − 1.73·11-s + (−0.802 + 0.802i)13-s + (0.311 − 1.10i)15-s + (0.517 + 0.517i)17-s + 0.502·19-s + (0.799 − 0.830i)21-s + (−1.00 − 1.00i)23-s + (0.520 − 0.853i)25-s + (0.546 + 0.546i)27-s + 0.964i·29-s − 1.20i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.690 + 0.723i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.690 + 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.690 + 0.723i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (153, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.690 + 0.723i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.68725 - 0.721993i\)
\(L(\frac12)\) \(\approx\) \(1.68725 - 0.721993i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.94 + 1.09i)T \)
7 \( 1 + (-2.64 + 0.0496i)T \)
good3 \( 1 + (-1.41 + 1.41i)T - 3iT^{2} \)
11 \( 1 + 5.75T + 11T^{2} \)
13 \( 1 + (2.89 - 2.89i)T - 13iT^{2} \)
17 \( 1 + (-2.13 - 2.13i)T + 17iT^{2} \)
19 \( 1 - 2.18T + 19T^{2} \)
23 \( 1 + (4.79 + 4.79i)T + 23iT^{2} \)
29 \( 1 - 5.19iT - 29T^{2} \)
31 \( 1 + 6.68iT - 31T^{2} \)
37 \( 1 + (6.50 - 6.50i)T - 37iT^{2} \)
41 \( 1 + 0.846iT - 41T^{2} \)
43 \( 1 + (-2.68 - 2.68i)T + 43iT^{2} \)
47 \( 1 + (4.55 + 4.55i)T + 47iT^{2} \)
53 \( 1 + (0.750 + 0.750i)T + 53iT^{2} \)
59 \( 1 - 4.25T + 59T^{2} \)
61 \( 1 - 7.62iT - 61T^{2} \)
67 \( 1 + (2.14 - 2.14i)T - 67iT^{2} \)
71 \( 1 + 2.44T + 71T^{2} \)
73 \( 1 + (-3.51 + 3.51i)T - 73iT^{2} \)
79 \( 1 - 1.15iT - 79T^{2} \)
83 \( 1 + (11.1 - 11.1i)T - 83iT^{2} \)
89 \( 1 - 5.57T + 89T^{2} \)
97 \( 1 + (-5.66 - 5.66i)T + 97iT^{2} \)
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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

−12.01772038574595526121536207305, −10.63983215680862275535891066601, −9.843236970282292496880820377468, −8.539128390135092184948177677261, −8.033166831610661659586003429354, −7.08200956705978380273063715196, −5.57829333457856115370797016344, −4.70545550581049037944147154259, −2.59820092832504847363748370508, −1.74963597756468648930505566465, 2.29747992236616892058233565272, 3.28069335904096456131748387672, 4.95376349855858278918432200240, 5.60634978570887739156145942555, 7.44548071755677510161148444573, 8.091054878342629184640549409165, 9.321818978277065652663741167120, 10.14424472180510958229773097062, 10.60930125613026738636144549328, 11.91899749584562095847232261902

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