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Properties

Label 2-4400-5.4-c1-0-44
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 + 0.561i·7-s − 3.56·9-s − 11-s − 5.12i·13-s + 1.43i·17-s + 6.56·19-s − 1.43·21-s − 1.12i·23-s − 1.43i·27-s + 4.56·29-s + 3.68·31-s − 2.56i·33-s − 10.8i·37-s + 13.1·39-s + ⋯
L(s)  = 1  + 1.47i·3-s + 0.212i·7-s − 1.18·9-s − 0.301·11-s − 1.42i·13-s + 0.348i·17-s + 1.50·19-s − 0.313·21-s − 0.234i·23-s − 0.276i·27-s + 0.847·29-s + 0.661·31-s − 0.445i·33-s − 1.77i·37-s + 2.10·39-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

\(L(1)\) \(\approx\) \(1.917486102\)
\(L(\frac12)\) \(\approx\) \(1.917486102\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.561iT - 7T^{2} \)
13 \( 1 + 5.12iT - 13T^{2} \)
17 \( 1 - 1.43iT - 17T^{2} \)
19 \( 1 - 6.56T + 19T^{2} \)
23 \( 1 + 1.12iT - 23T^{2} \)
29 \( 1 - 4.56T + 29T^{2} \)
31 \( 1 - 3.68T + 31T^{2} \)
37 \( 1 + 10.8iT - 37T^{2} \)
41 \( 1 + 10T + 41T^{2} \)
43 \( 1 + 3.12iT - 43T^{2} \)
47 \( 1 + 1.12iT - 47T^{2} \)
53 \( 1 - 8.56iT - 53T^{2} \)
59 \( 1 - 11.3T + 59T^{2} \)
61 \( 1 - 0.561T + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 - 6.56T + 71T^{2} \)
73 \( 1 + 13.1iT - 73T^{2} \)
79 \( 1 - 9.12T + 79T^{2} \)
83 \( 1 - 10iT - 83T^{2} \)
89 \( 1 - 10.8T + 89T^{2} \)
97 \( 1 - 8.87iT - 97T^{2} \)
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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

−8.581707109290450948884615036263, −7.935430044330211258871617103078, −7.12387857599123135181578963179, −5.98886176595513959229814323084, −5.31512947293522054808815254387, −4.91929116527584600253103450214, −3.84481089148177753545168750774, −3.29392126450786899591487416374, −2.44354618291379350056410987913, −0.76393843353810014569136222422, 0.834880775036371828231252068920, 1.63699770702775667378502875092, 2.55288805715954989668980148497, 3.44449017449463466322699420227, 4.61673153698482181999119608965, 5.34120933677842221408224797549, 6.38505332724044090218286387609, 6.83043552941864889707054707250, 7.35566570085477413900764745539, 8.153680633304313077444330902492

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