Properties

Label 2-4400-5.4-c1-0-65
Degree $2$
Conductor $4400$
Sign $0.447 + 0.894i$
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  + 2.56i·3-s − 2.56i·7-s − 3.56·9-s − 11-s + 2i·13-s + 0.561i·17-s + 2.56·19-s + 6.56·21-s − 5.12i·23-s − 1.43i·27-s − 9.68·29-s − 6.56·31-s − 2.56i·33-s + 5.68i·37-s − 5.12·39-s + ⋯
L(s)  = 1  + 1.47i·3-s − 0.968i·7-s − 1.18·9-s − 0.301·11-s + 0.554i·13-s + 0.136i·17-s + 0.587·19-s + 1.43·21-s − 1.06i·23-s − 0.276i·27-s − 1.79·29-s − 1.17·31-s − 0.445i·33-s + 0.934i·37-s − 0.820·39-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(4400\)    =    \(2^{4} \cdot 5^{2} \cdot 11\)
Sign: $0.447 + 0.894i$
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: \((2,\ 4400,\ (\ :1/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7615732443\)
\(L(\frac12)\) \(\approx\) \(0.7615732443\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.56iT - 3T^{2} \)
7 \( 1 + 2.56iT - 7T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 - 0.561iT - 17T^{2} \)
19 \( 1 - 2.56T + 19T^{2} \)
23 \( 1 + 5.12iT - 23T^{2} \)
29 \( 1 + 9.68T + 29T^{2} \)
31 \( 1 + 6.56T + 31T^{2} \)
37 \( 1 - 5.68iT - 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + 10.2iT - 43T^{2} \)
47 \( 1 + 13.1iT - 47T^{2} \)
53 \( 1 - 4.56iT - 53T^{2} \)
59 \( 1 - 1.12T + 59T^{2} \)
61 \( 1 - 2.31T + 61T^{2} \)
67 \( 1 - 6.24iT - 67T^{2} \)
71 \( 1 + 3.68T + 71T^{2} \)
73 \( 1 + 2iT - 73T^{2} \)
79 \( 1 + 15.3T + 79T^{2} \)
83 \( 1 + 5.12iT - 83T^{2} \)
89 \( 1 - 12.5T + 89T^{2} \)
97 \( 1 + 7.12iT - 97T^{2} \)
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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

−8.481903778283866838241340102626, −7.37375530626784333813826250266, −6.96713589044045176645298248924, −5.73043128293770188095925728330, −5.20210533202993948726125663401, −4.24164060576108072551550875449, −3.91803491941726052986291776675, −3.07115207355862851677081857060, −1.80867841030892429950969858239, −0.21531089094521229089344144852, 1.17765031794342621784151060168, 2.05124672288470713072205294571, 2.79817130821994376818001268976, 3.72827192439604540352306841436, 5.12666041615254361737532666183, 5.72370614529849194671141234188, 6.21072393731730160942941452721, 7.33908637874224176198679453377, 7.56503001018456168706660553695, 8.241510184942919508912097308787

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