Properties

Label 2-4410-1.1-c1-0-17
Degree $2$
Conductor $4410$
Sign $1$
Analytic cond. $35.2140$
Root an. cond. $5.93414$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 5-s + 8-s − 10-s − 3.41·11-s + 1.17·13-s + 16-s + 3.41·17-s + 4.82·19-s − 20-s − 3.41·22-s + 3.65·23-s + 25-s + 1.17·26-s − 5.41·29-s + 7.41·31-s + 32-s + 3.41·34-s − 3.41·37-s + 4.82·38-s − 40-s − 3.65·41-s − 7.07·43-s − 3.41·44-s + 3.65·46-s + 2.58·47-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s − 0.447·5-s + 0.353·8-s − 0.316·10-s − 1.02·11-s + 0.324·13-s + 0.250·16-s + 0.828·17-s + 1.10·19-s − 0.223·20-s − 0.727·22-s + 0.762·23-s + 0.200·25-s + 0.229·26-s − 1.00·29-s + 1.33·31-s + 0.176·32-s + 0.585·34-s − 0.561·37-s + 0.783·38-s − 0.158·40-s − 0.571·41-s − 1.07·43-s − 0.514·44-s + 0.539·46-s + 0.377·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4410\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(35.2140\)
Root analytic conductor: \(5.93414\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4410} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4410,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.808136947\)
\(L(\frac12)\) \(\approx\) \(2.808136947\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
5 \( 1 + T \)
7 \( 1 \)
good11 \( 1 + 3.41T + 11T^{2} \)
13 \( 1 - 1.17T + 13T^{2} \)
17 \( 1 - 3.41T + 17T^{2} \)
19 \( 1 - 4.82T + 19T^{2} \)
23 \( 1 - 3.65T + 23T^{2} \)
29 \( 1 + 5.41T + 29T^{2} \)
31 \( 1 - 7.41T + 31T^{2} \)
37 \( 1 + 3.41T + 37T^{2} \)
41 \( 1 + 3.65T + 41T^{2} \)
43 \( 1 + 7.07T + 43T^{2} \)
47 \( 1 - 2.58T + 47T^{2} \)
53 \( 1 + 6.48T + 53T^{2} \)
59 \( 1 + 0.828T + 59T^{2} \)
61 \( 1 + 1.65T + 61T^{2} \)
67 \( 1 - 5.89T + 67T^{2} \)
71 \( 1 - 10.4T + 71T^{2} \)
73 \( 1 - 12.8T + 73T^{2} \)
79 \( 1 - 13.6T + 79T^{2} \)
83 \( 1 - 0.485T + 83T^{2} \)
89 \( 1 - 14.4T + 89T^{2} \)
97 \( 1 - 14.4T + 97T^{2} \)
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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

−8.040264317826222713448324677190, −7.70046186847754064570118579352, −6.87595568515737267107206846034, −6.08431808317092472635219522169, −5.12704258308494886166040579866, −4.90931216342004858491689244602, −3.56138586346781589082217564499, −3.24010507312999346875706925640, −2.13105282539703761973567226224, −0.853874032457850781213992552000, 0.853874032457850781213992552000, 2.13105282539703761973567226224, 3.24010507312999346875706925640, 3.56138586346781589082217564499, 4.90931216342004858491689244602, 5.12704258308494886166040579866, 6.08431808317092472635219522169, 6.87595568515737267107206846034, 7.70046186847754064570118579352, 8.040264317826222713448324677190

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