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Properties

Label 2-4400-5.4-c1-0-3
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  + 3i·3-s − i·7-s − 6·9-s + 11-s + 6i·13-s + 3i·17-s − 5·19-s + 3·21-s − 2i·23-s − 9i·27-s + 5·29-s − 5·31-s + 3i·33-s − i·37-s − 18·39-s + ⋯
L(s)  = 1  + 1.73i·3-s − 0.377i·7-s − 2·9-s + 0.301·11-s + 1.66i·13-s + 0.727i·17-s − 1.14·19-s + 0.654·21-s − 0.417i·23-s − 1.73i·27-s + 0.928·29-s − 0.898·31-s + 0.522i·33-s − 0.164i·37-s − 2.88·39-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

\(L(1)\) \(\approx\) \(0.7426758511\)
\(L(\frac12)\) \(\approx\) \(0.7426758511\)
\(L(\frac{3}{2})\) 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 \( 1 - 3iT - 3T^{2} \)
7 \( 1 + iT - 7T^{2} \)
13 \( 1 - 6iT - 13T^{2} \)
17 \( 1 - 3iT - 17T^{2} \)
19 \( 1 + 5T + 19T^{2} \)
23 \( 1 + 2iT - 23T^{2} \)
29 \( 1 - 5T + 29T^{2} \)
31 \( 1 + 5T + 31T^{2} \)
37 \( 1 + iT - 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 - 12iT - 43T^{2} \)
47 \( 1 - 2iT - 47T^{2} \)
53 \( 1 - 13iT - 53T^{2} \)
59 \( 1 - 2T + 59T^{2} \)
61 \( 1 - T + 61T^{2} \)
67 \( 1 + 16iT - 67T^{2} \)
71 \( 1 + 15T + 71T^{2} \)
73 \( 1 + 10iT - 73T^{2} \)
79 \( 1 - 2T + 79T^{2} \)
83 \( 1 + 14iT - 83T^{2} \)
89 \( 1 + 9T + 89T^{2} \)
97 \( 1 + 16iT - 97T^{2} \)
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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

−8.964722067397250987354659572850, −8.491947578471078050859196069926, −7.43155566402305188568415506118, −6.41510792573301496361869949425, −5.97150450517369586504814704358, −4.70451107946103398698356276433, −4.41028586967134092555097719726, −3.82624810146635820863960409440, −2.86502114178236121924022417408, −1.68887215800511265929924564329, 0.21166504532439318488606333166, 1.17609305405654517256820405933, 2.25368052621820850430915663428, 2.83931088141210393407492867848, 3.90371787984134501467885209437, 5.42183973089455666128726808786, 5.56566807120441745495240698865, 6.70973625946219595977389806709, 6.99349658125784556360245309434, 7.88736066081743336438639521403

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