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Properties

Label 2-4410-21.20-c1-0-14
Degree $2$
Conductor $4410$
Sign $0.981 - 0.192i$
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  − i·2-s − 4-s − 5-s + i·8-s + i·10-s − 3.84i·11-s − 0.0273i·13-s + 16-s − 3.71·17-s + 2.94i·19-s + 20-s − 3.84·22-s + 9.05i·23-s + 25-s − 0.0273·26-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s − 0.447·5-s + 0.353i·8-s + 0.316i·10-s − 1.16i·11-s − 0.00758i·13-s + 0.250·16-s − 0.899·17-s + 0.675i·19-s + 0.223·20-s − 0.820·22-s + 1.88i·23-s + 0.200·25-s − 0.00536·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.192i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 - 0.192i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4410\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.981 - 0.192i$
Analytic conductor: \(35.2140\)
Root analytic conductor: \(5.93414\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4410} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4410,\ (\ :1/2),\ 0.981 - 0.192i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.074512421\)
\(L(\frac12)\) \(\approx\) \(1.074512421\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 \)
5 \( 1 + T \)
7 \( 1 \)
good11 \( 1 + 3.84iT - 11T^{2} \)
13 \( 1 + 0.0273iT - 13T^{2} \)
17 \( 1 + 3.71T + 17T^{2} \)
19 \( 1 - 2.94iT - 19T^{2} \)
23 \( 1 - 9.05iT - 23T^{2} \)
29 \( 1 + 9.28iT - 29T^{2} \)
31 \( 1 - 2.20iT - 31T^{2} \)
37 \( 1 - 3.00T + 37T^{2} \)
41 \( 1 + 8.61T + 41T^{2} \)
43 \( 1 + 3.28T + 43T^{2} \)
47 \( 1 - 10.7T + 47T^{2} \)
53 \( 1 - 4.84iT - 53T^{2} \)
59 \( 1 + 4.04T + 59T^{2} \)
61 \( 1 - 5.88iT - 61T^{2} \)
67 \( 1 - 10.6T + 67T^{2} \)
71 \( 1 + 1.14iT - 71T^{2} \)
73 \( 1 - 7.67iT - 73T^{2} \)
79 \( 1 + 5.65T + 79T^{2} \)
83 \( 1 - 3.22T + 83T^{2} \)
89 \( 1 + 7.28T + 89T^{2} \)
97 \( 1 + 1.39iT - 97T^{2} \)
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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

−8.407224600007500198150672455929, −7.86695101097464442620546860136, −7.00903337042311214485374641230, −5.99834100218044283277622562551, −5.46132858635678966283254958592, −4.38771476284564037237036321695, −3.73308857453910993529617872693, −3.02783871320143849565770411778, −1.98870849990091482803563009914, −0.862218762955715603395576579062, 0.38844781359066976175666327225, 1.91157660460685744227134326656, 2.93231896218991279077269239094, 4.07358424016658064752624816221, 4.67073311639001941517520875517, 5.24164721493265343813517046919, 6.47287461685640893798213025536, 6.84017320667445356333821452271, 7.46366802023338359015584730016, 8.384645712264941570747978711795

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