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Properties

Label 2-4410-105.104-c1-0-44
Degree $2$
Conductor $4410$
Sign $0.952 + 0.305i$
Analytic cond. $35.2140$
Root an. cond. $5.93414$
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-s + 4-s + (−2.00 + 0.997i)5-s + 8-s + (−2.00 + 0.997i)10-s + 0.311i·11-s − 1.62·13-s + 16-s − 1.60i·17-s − 4.54i·19-s + (−2.00 + 0.997i)20-s + 0.311i·22-s + 4.42·23-s + (3.00 − 3.99i)25-s − 1.62·26-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + (−0.894 + 0.446i)5-s + 0.353·8-s + (−0.632 + 0.315i)10-s + 0.0939i·11-s − 0.451·13-s + 0.250·16-s − 0.390i·17-s − 1.04i·19-s + (−0.447 + 0.223i)20-s + 0.0664i·22-s + 0.922·23-s + (0.601 − 0.798i)25-s − 0.319·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 + 0.305i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.952 + 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4410\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.952 + 0.305i$
Analytic conductor: \(35.2140\)
Root analytic conductor: \(5.93414\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4410} (4409, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4410,\ (\ :1/2),\ 0.952 + 0.305i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.373995432\)
\(L(\frac12)\) \(\approx\) \(2.373995432\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.00 - 0.997i)T \)
7 \( 1 \)
good11 \( 1 - 0.311iT - 11T^{2} \)
13 \( 1 + 1.62T + 13T^{2} \)
17 \( 1 + 1.60iT - 17T^{2} \)
19 \( 1 + 4.54iT - 19T^{2} \)
23 \( 1 - 4.42T + 23T^{2} \)
29 \( 1 - 1.79iT - 29T^{2} \)
31 \( 1 - 0.415iT - 31T^{2} \)
37 \( 1 - 2.16iT - 37T^{2} \)
41 \( 1 + 3.39T + 41T^{2} \)
43 \( 1 - 0.812iT - 43T^{2} \)
47 \( 1 - 0.316iT - 47T^{2} \)
53 \( 1 - 11.5T + 53T^{2} \)
59 \( 1 - 10.0T + 59T^{2} \)
61 \( 1 + 9.94iT - 61T^{2} \)
67 \( 1 + 14.8iT - 67T^{2} \)
71 \( 1 - 2.38iT - 71T^{2} \)
73 \( 1 + 6.13T + 73T^{2} \)
79 \( 1 - 9.32T + 79T^{2} \)
83 \( 1 - 9.49iT - 83T^{2} \)
89 \( 1 + 10.0T + 89T^{2} \)
97 \( 1 - 12.7T + 97T^{2} \)
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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

−8.218410252396148921253395427203, −7.25337241054400328244016931695, −7.03913046252138108864024268838, −6.19946217849533990476346174424, −5.05861003327562594666532757314, −4.71688797272662877067355500700, −3.69399297206925901795816540162, −3.04056075070506407450583113878, −2.21472703092852362249307832522, −0.66607573569027429563945121345, 0.900252048609788913006232160761, 2.12072577886203031134241148716, 3.20997089701658948833095809663, 3.90626158274144207973776781887, 4.57359149202350991773416006570, 5.37977654807155220009264355740, 6.02459042763243156284949094460, 7.11190939821105171529941977524, 7.45241149317614933386795255485, 8.414826670665422906564427884111

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