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Properties

Label 2-2240-280.139-c1-0-9
Degree $2$
Conductor $2240$
Sign $-0.874 - 0.485i$
Analytic cond. $17.8864$
Root an. cond. $4.22924$
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  − 2.97·3-s + (−2.14 + 0.615i)5-s − 2.64·7-s + 5.82·9-s + 7.17i·13-s + (6.38 − 1.82i)15-s − 6.81i·19-s + 7.86·21-s + 7.48·23-s + (4.24 − 2.64i)25-s − 8.40·27-s + (5.68 − 1.62i)35-s − 21.3i·39-s + (−12.5 + 3.58i)45-s + 7.00·49-s + ⋯
L(s)  = 1  − 1.71·3-s + (−0.961 + 0.275i)5-s − 0.999·7-s + 1.94·9-s + 1.98i·13-s + (1.64 − 0.472i)15-s − 1.56i·19-s + 1.71·21-s + 1.56·23-s + (0.848 − 0.529i)25-s − 1.61·27-s + (0.961 − 0.275i)35-s − 3.41i·39-s + (−1.86 + 0.534i)45-s + 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.874 - 0.485i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.874 - 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $-0.874 - 0.485i$
Analytic conductor: \(17.8864\)
Root analytic conductor: \(4.22924\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2240} (1119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2240,\ (\ :1/2),\ -0.874 - 0.485i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2772315113\)
\(L(\frac12)\) \(\approx\) \(0.2772315113\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.14 - 0.615i)T \)
7 \( 1 + 2.64T \)
good3 \( 1 + 2.97T + 3T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 7.17iT - 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 6.81iT - 19T^{2} \)
23 \( 1 - 7.48T + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 13.9iT - 59T^{2} \)
61 \( 1 - 11.4T + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 5.65iT - 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 - 16.9iT - 79T^{2} \)
83 \( 1 + 13.8T + 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−9.351110346546565832869467050916, −8.817669044361692471023579519093, −7.30841272249885173478743103222, −6.79776479112425753579890727336, −6.55747251232771638070175466516, −5.38577780136230677468135614728, −4.57457141807129112817743367401, −3.96519773348299287358824045411, −2.69969063927151262840021220553, −0.969931827526012026402293686003, 0.19230658272492545818410625937, 1.04310509717718310794975407130, 3.10374822408331634109803465561, 3.84301467226114177626323047452, 4.97209647965266018282097954295, 5.52489079216937152834837772194, 6.23913342589545281109305962173, 7.07199626924104639429601647490, 7.75600050834634994113819091841, 8.617367717452062259075599404444

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