Properties

Label 2-4410-1.1-c1-0-14
Degree $2$
Conductor $4410$
Sign $1$
Analytic cond. $35.2140$
Root an. cond. $5.93414$
Motivic weight $1$
Arithmetic yes
Rational yes
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 − 6·11-s + 6·13-s + 16-s − 4·19-s − 20-s − 6·22-s + 25-s + 6·26-s + 8·29-s + 2·31-s + 32-s + 4·37-s − 4·38-s − 40-s + 10·41-s − 6·43-s − 6·44-s + 2·47-s + 50-s + 6·52-s − 10·53-s + 6·55-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.353·8-s − 0.316·10-s − 1.80·11-s + 1.66·13-s + 1/4·16-s − 0.917·19-s − 0.223·20-s − 1.27·22-s + 1/5·25-s + 1.17·26-s + 1.48·29-s + 0.359·31-s + 0.176·32-s + 0.657·37-s − 0.648·38-s − 0.158·40-s + 1.56·41-s − 0.914·43-s − 0.904·44-s + 0.291·47-s + 0.141·50-s + 0.832·52-s − 1.37·53-s + 0.809·55-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: yes
Arithmetic: yes
Character: $\chi_{4410} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4410,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.637476455\)
\(L(\frac12)\) \(\approx\) \(2.637476455\)
\(L(\frac{3}{2})\) 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 \( 1 + 6 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−8.160279983269438934100881939312, −7.79170468641312060938137918332, −6.68867036961452943989019141336, −6.16236649160420655702633062986, −5.33348176211912876532962783406, −4.61091460088952365750923964785, −3.84585699504002316995217387933, −2.99348228626267769382056994446, −2.22774537907891883805264475213, −0.814028269383723992773983515515, 0.814028269383723992773983515515, 2.22774537907891883805264475213, 2.99348228626267769382056994446, 3.84585699504002316995217387933, 4.61091460088952365750923964785, 5.33348176211912876532962783406, 6.16236649160420655702633062986, 6.68867036961452943989019141336, 7.79170468641312060938137918332, 8.160279983269438934100881939312

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