Properties

Label 2-4400-5.4-c1-0-35
Degree $2$
Conductor $4400$
Sign $-0.894 - 0.447i$
Analytic cond. $35.1341$
Root an. cond. $5.92740$
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  + 1.30i·3-s + 4.30i·7-s + 1.30·9-s + 11-s + 5i·13-s + 3.90i·17-s − 19-s − 5.60·21-s + 3.69i·23-s + 5.60i·27-s + 9.90·29-s + 4.21·31-s + 1.30i·33-s − 9.60i·37-s − 6.51·39-s + ⋯
L(s)  = 1  + 0.752i·3-s + 1.62i·7-s + 0.434·9-s + 0.301·11-s + 1.38i·13-s + 0.947i·17-s − 0.229·19-s − 1.22·21-s + 0.770i·23-s + 1.07i·27-s + 1.83·29-s + 0.756·31-s + 0.226i·33-s − 1.57i·37-s − 1.04·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4400\)    =    \(2^{4} \cdot 5^{2} \cdot 11\)
Sign: $-0.894 - 0.447i$
Analytic conductor: \(35.1341\)
Root analytic conductor: \(5.92740\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4400} (4049, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4400,\ (\ :1/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.071073901\)
\(L(\frac12)\) \(\approx\) \(2.071073901\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11 \( 1 - T \)
good3 \( 1 - 1.30iT - 3T^{2} \)
7 \( 1 - 4.30iT - 7T^{2} \)
13 \( 1 - 5iT - 13T^{2} \)
17 \( 1 - 3.90iT - 17T^{2} \)
19 \( 1 + T + 19T^{2} \)
23 \( 1 - 3.69iT - 23T^{2} \)
29 \( 1 - 9.90T + 29T^{2} \)
31 \( 1 - 4.21T + 31T^{2} \)
37 \( 1 + 9.60iT - 37T^{2} \)
41 \( 1 - 1.60T + 41T^{2} \)
43 \( 1 - 7.21iT - 43T^{2} \)
47 \( 1 + 3iT - 47T^{2} \)
53 \( 1 - 2.30iT - 53T^{2} \)
59 \( 1 - 0.211T + 59T^{2} \)
61 \( 1 - 2.90T + 61T^{2} \)
67 \( 1 + 4iT - 67T^{2} \)
71 \( 1 + 4.60T + 71T^{2} \)
73 \( 1 - 2.90iT - 73T^{2} \)
79 \( 1 + 0.0916T + 79T^{2} \)
83 \( 1 + 14.5iT - 83T^{2} \)
89 \( 1 + 5.30T + 89T^{2} \)
97 \( 1 + 11.6iT - 97T^{2} \)
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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

−8.854349467420096433923769629788, −8.184706038497842868937869257146, −7.13226757602496250769499103254, −6.31913508663060298860209540310, −5.79541986940834833416421878507, −4.79026280665010030185415953282, −4.29948723498312921359661198126, −3.37497581786662752671429438514, −2.34768618212127408704412799144, −1.52648357535240705285244676822, 0.68386610627091710063430171487, 1.10382496678185013011507823931, 2.52173082535976043270798528711, 3.37308117511859779474210871641, 4.39015237267620016071973132426, 4.87475472329821282395741608023, 6.11044413553715349916126183037, 6.83269815134482350696774564533, 7.17468786208329908313001531958, 8.039561510314826712283770928498

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