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Properties

Label 2-4410-105.104-c1-0-53
Degree $2$
Conductor $4410$
Sign $0.492 + 0.870i$
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 + (2.12 − 0.705i)5-s − 8-s + (−2.12 + 0.705i)10-s + 3.38i·11-s + 0.920·13-s + 16-s − 7.01i·17-s + 0.709i·19-s + (2.12 − 0.705i)20-s − 3.38i·22-s + 2.11·23-s + (4.00 − 2.99i)25-s − 0.920·26-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + (0.948 − 0.315i)5-s − 0.353·8-s + (−0.671 + 0.223i)10-s + 1.01i·11-s + 0.255·13-s + 0.250·16-s − 1.70i·17-s + 0.162i·19-s + (0.474 − 0.157i)20-s − 0.720i·22-s + 0.441·23-s + (0.800 − 0.598i)25-s − 0.180·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.492 + 0.870i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.492 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4410\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.492 + 0.870i$
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: \((2,\ 4410,\ (\ :1/2),\ 0.492 + 0.870i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.540727821\)
\(L(\frac12)\) \(\approx\) \(1.540727821\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.12 + 0.705i)T \)
7 \( 1 \)
good11 \( 1 - 3.38iT - 11T^{2} \)
13 \( 1 - 0.920T + 13T^{2} \)
17 \( 1 + 7.01iT - 17T^{2} \)
19 \( 1 - 0.709iT - 19T^{2} \)
23 \( 1 - 2.11T + 23T^{2} \)
29 \( 1 - 0.0694iT - 29T^{2} \)
31 \( 1 + 4.09iT - 31T^{2} \)
37 \( 1 - 2.81iT - 37T^{2} \)
41 \( 1 + 10.8T + 41T^{2} \)
43 \( 1 + 7.35iT - 43T^{2} \)
47 \( 1 + 10.9iT - 47T^{2} \)
53 \( 1 + 2.79T + 53T^{2} \)
59 \( 1 - 5.80T + 59T^{2} \)
61 \( 1 - 5.18iT - 61T^{2} \)
67 \( 1 - 6.80iT - 67T^{2} \)
71 \( 1 + 13.9iT - 71T^{2} \)
73 \( 1 - 6.18T + 73T^{2} \)
79 \( 1 - 4.57T + 79T^{2} \)
83 \( 1 + 8.39iT - 83T^{2} \)
89 \( 1 + 0.636T + 89T^{2} \)
97 \( 1 - 3.00T + 97T^{2} \)
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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

−8.457175015520339952479904860153, −7.37820333120946646836020701206, −6.97578797976340191076272066218, −6.16991535738518405139379148052, −5.22891264891192018233488463650, −4.76521949165477049184692665757, −3.46893266735947059557497269976, −2.42823506904221465122440299655, −1.76855659770168635859757497943, −0.58950628893477712551352882371, 1.09948128814463796585572295042, 1.91230723322448864875172530505, 2.96934818710240857474205602803, 3.66076529720588082892595279806, 4.94129730380942991322724698741, 5.82876383950570832997200650577, 6.31430856858838878532669438054, 6.92471533874565870297247453828, 7.994827722263059729275973356751, 8.501876462201783555699497783092

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