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Properties

Label 2-2240-280.139-c1-0-28
Degree $2$
Conductor $2240$
Sign $0.689 - 0.724i$
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

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(1.176693027\)
\(L(\frac12)\) \(\approx\) \(1.176693027\)
\(L(\frac{3}{2})\) 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

−8.858623072071207989475324431209, −8.285393564745044335452173111742, −7.84658458254102316497438013273, −6.99105687321984482321169510500, −5.71407522090089534376827132483, −5.19364768163309677257843705971, −4.34430602213419741575489149680, −3.49376772648709768518113918282, −2.24272246280163187360359375747, −1.03020638287261687169096720594, 0.48025296122531240236736813808, 2.38140966551746300664480787465, 2.90786703816612024534591529841, 3.98076673566125440745897077701, 5.14091487849510709009179341940, 5.63181965166586710395126460552, 6.60052511076057799503065904744, 7.69349227054962840031149238900, 8.146552759251016842913038424288, 8.511704433130563563641402657935

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