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Properties

Label 2-2720-17.16-c1-0-44
Degree $2$
Conductor $2720$
Sign $0.860 + 0.508i$
Analytic cond. $21.7193$
Root an. cond. $4.66039$
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  + 1.67i·3-s + i·5-s − 4.87i·7-s + 0.199·9-s − 1.43i·11-s − 3.49·13-s − 1.67·15-s + (−2.09 + 3.54i)17-s + 1.25·19-s + 8.15·21-s + 4.27i·23-s − 25-s + 5.35i·27-s + 2.30i·29-s − 8.16i·31-s + ⋯
L(s)  = 1  + 0.966i·3-s + 0.447i·5-s − 1.84i·7-s + 0.0664·9-s − 0.432i·11-s − 0.969·13-s − 0.432·15-s + (−0.508 + 0.860i)17-s + 0.287·19-s + 1.77·21-s + 0.891i·23-s − 0.200·25-s + 1.03i·27-s + 0.427i·29-s − 1.46i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 + 0.508i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.860 + 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2720\)    =    \(2^{5} \cdot 5 \cdot 17\)
Sign: $0.860 + 0.508i$
Analytic conductor: \(21.7193\)
Root analytic conductor: \(4.66039\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2720} (1121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2720,\ (\ :1/2),\ 0.860 + 0.508i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.521134790\)
\(L(\frac12)\) \(\approx\) \(1.521134790\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
17 \( 1 + (2.09 - 3.54i)T \)
good3 \( 1 - 1.67iT - 3T^{2} \)
7 \( 1 + 4.87iT - 7T^{2} \)
11 \( 1 + 1.43iT - 11T^{2} \)
13 \( 1 + 3.49T + 13T^{2} \)
19 \( 1 - 1.25T + 19T^{2} \)
23 \( 1 - 4.27iT - 23T^{2} \)
29 \( 1 - 2.30iT - 29T^{2} \)
31 \( 1 + 8.16iT - 31T^{2} \)
37 \( 1 + 6.75iT - 37T^{2} \)
41 \( 1 + 6.23iT - 41T^{2} \)
43 \( 1 - 10.8T + 43T^{2} \)
47 \( 1 - 5.26T + 47T^{2} \)
53 \( 1 - 5.11T + 53T^{2} \)
59 \( 1 - 7.32T + 59T^{2} \)
61 \( 1 + 14.0iT - 61T^{2} \)
67 \( 1 - 7.18T + 67T^{2} \)
71 \( 1 + 11.1iT - 71T^{2} \)
73 \( 1 + 15.1iT - 73T^{2} \)
79 \( 1 + 10.5iT - 79T^{2} \)
83 \( 1 - 5.75T + 83T^{2} \)
89 \( 1 + 5.87T + 89T^{2} \)
97 \( 1 - 0.696iT - 97T^{2} \)
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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

−9.029846767806673016725758853179, −7.63531206865572717232385553800, −7.49651093379324703597680607800, −6.57212171618504595171674659135, −5.56707596343834528253335153480, −4.60316962388728404612100940005, −3.92037587468389774328322579119, −3.49153805482430086651204440036, −2.05122076079071418179833524182, −0.55077685393170504070492891252, 1.11021306691179762533462726365, 2.42457349775544429871396314779, 2.58541289573999708890696554377, 4.36309992561487518060180600494, 5.11454631366646905737299282604, 5.80446016427373577786530109168, 6.75460539693920083406248349493, 7.25380969750267246068543861452, 8.266020825847877977734743422603, 8.726239211717153102085357312634

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