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Properties

Label 2-2240-280.139-c1-0-44
Degree $2$
Conductor $2240$
Sign $0.724 + 0.689i$
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  + (−1.41 − 1.73i)5-s + (−2.44 + i)7-s − 3·9-s + 3.46·11-s + 3.46i·13-s + 4·17-s − 2.82i·19-s + 4.89·23-s + (−0.999 + 4.89i)25-s + 6.92i·29-s − 9.79·31-s + (5.19 + 2.82i)35-s − 5.65·37-s − 9.79i·41-s − 2.82i·43-s + ⋯
L(s)  = 1  + (−0.632 − 0.774i)5-s + (−0.925 + 0.377i)7-s − 9-s + 1.04·11-s + 0.960i·13-s + 0.970·17-s − 0.648i·19-s + 1.02·23-s + (−0.199 + 0.979i)25-s + 1.28i·29-s − 1.75·31-s + (0.878 + 0.478i)35-s − 0.929·37-s − 1.53i·41-s − 0.431i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.724 + 0.689i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.724 + 0.689i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $0.724 + 0.689i$
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: \((2,\ 2240,\ (\ :1/2),\ 0.724 + 0.689i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.157299411\)
\(L(\frac12)\) \(\approx\) \(1.157299411\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.41 + 1.73i)T \)
7 \( 1 + (2.44 - i)T \)
good3 \( 1 + 3T^{2} \)
11 \( 1 - 3.46T + 11T^{2} \)
13 \( 1 - 3.46iT - 13T^{2} \)
17 \( 1 - 4T + 17T^{2} \)
19 \( 1 + 2.82iT - 19T^{2} \)
23 \( 1 - 4.89T + 23T^{2} \)
29 \( 1 - 6.92iT - 29T^{2} \)
31 \( 1 + 9.79T + 31T^{2} \)
37 \( 1 + 5.65T + 37T^{2} \)
41 \( 1 + 9.79iT - 41T^{2} \)
43 \( 1 + 2.82iT - 43T^{2} \)
47 \( 1 + 10iT - 47T^{2} \)
53 \( 1 - 11.3T + 53T^{2} \)
59 \( 1 + 8.48iT - 59T^{2} \)
61 \( 1 - 14.1T + 61T^{2} \)
67 \( 1 - 14.1iT - 67T^{2} \)
71 \( 1 + 4iT - 71T^{2} \)
73 \( 1 - 12T + 73T^{2} \)
79 \( 1 + 12iT - 79T^{2} \)
83 \( 1 - 13.8T + 83T^{2} \)
89 \( 1 + 9.79iT - 89T^{2} \)
97 \( 1 - 4T + 97T^{2} \)
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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

−8.993750845488136238360897208914, −8.524540982895383011177832020178, −7.12233524899989504757009056341, −6.86878570467124974396124779315, −5.56741451995713408373778119824, −5.17521068047933745333156559207, −3.76794702418523235085956700844, −3.43203251232413174003467445541, −1.99919280279129028870866810718, −0.57174592993702278031863806562, 0.835041563737367004751385581521, 2.63145318787618838619906193945, 3.43615670721728573484216482868, 3.88223471161199756575087192487, 5.33247048137075113765300220659, 6.10567858962449259785726388964, 6.76431953821719652808071210221, 7.62298645135182671481037833691, 8.203471393836147137394215652864, 9.225595600957432234306408281006

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