Properties

Label 2-4410-21.20-c1-0-39
Degree $2$
Conductor $4410$
Sign $0.239 + 0.970i$
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 + 5.39i·11-s − 2.51i·13-s + 16-s − 4.49·17-s − 2.86i·19-s − 20-s − 5.39·22-s + 0.267i·23-s + 25-s + 2.51·26-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + 0.447·5-s − 0.353i·8-s + 0.316i·10-s + 1.62i·11-s − 0.698i·13-s + 0.250·16-s − 1.09·17-s − 0.656i·19-s − 0.223·20-s − 1.15·22-s + 0.0558i·23-s + 0.200·25-s + 0.493·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.239 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.239 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4410\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.239 + 0.970i$
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: \((2,\ 4410,\ (\ :1/2),\ 0.239 + 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6849827146\)
\(L(\frac12)\) \(\approx\) \(0.6849827146\)
\(L(\frac{3}{2})\) 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 \( 1 - 5.39iT - 11T^{2} \)
13 \( 1 + 2.51iT - 13T^{2} \)
17 \( 1 + 4.49T + 17T^{2} \)
19 \( 1 + 2.86iT - 19T^{2} \)
23 \( 1 - 0.267iT - 23T^{2} \)
29 \( 1 + 8.89iT - 29T^{2} \)
31 \( 1 + 4.82iT - 31T^{2} \)
37 \( 1 + 6.51T + 37T^{2} \)
41 \( 1 - 0.760T + 41T^{2} \)
43 \( 1 + 5.86T + 43T^{2} \)
47 \( 1 + 7.99T + 47T^{2} \)
53 \( 1 - 8.39iT - 53T^{2} \)
59 \( 1 + 12.6T + 59T^{2} \)
61 \( 1 + 2.62iT - 61T^{2} \)
67 \( 1 - 9.82T + 67T^{2} \)
71 \( 1 + 4.76iT - 71T^{2} \)
73 \( 1 + 11.6iT - 73T^{2} \)
79 \( 1 - 8.59T + 79T^{2} \)
83 \( 1 - 9.45T + 83T^{2} \)
89 \( 1 - 7.97T + 89T^{2} \)
97 \( 1 + 6.16iT - 97T^{2} \)
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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

−7.931860598616670691670900405356, −7.58113798575775953210876871326, −6.58168116905091621803984300511, −6.28565194158479058833206554246, −5.11226847367260277006131425519, −4.72979298856271288963838555487, −3.84110977620220671912999841911, −2.59521147542846716598627070179, −1.78700580336920732157079638011, −0.18813774500034311289542452344, 1.24232614829045070628351906316, 2.08299871999283245235007685728, 3.19399256958539664093000864186, 3.68387675236478587285646234069, 4.83531909201346583510749258380, 5.40173246468487851404374652398, 6.39179948618767201858329576469, 6.84266314063041287575063095153, 8.095796774785902736092903147699, 8.706549375349547149342160542163

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