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Properties

Label 2-4410-105.104-c1-0-25
Degree $2$
Conductor $4410$
Sign $0.656 - 0.754i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.656 - 0.754i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.656 - 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4410\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.656 - 0.754i$
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: \((2,\ 4410,\ (\ :1/2),\ 0.656 - 0.754i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.316684336\)
\(L(\frac12)\) \(\approx\) \(2.316684336\)
\(L(\frac{3}{2})\) 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

−8.236750854429400900822313242579, −7.80075080732186050352069041578, −6.92533638338196740971374633614, −6.05039845990134696034841011021, −5.74519063211541080947739182192, −4.44930541941841908845817338648, −3.93014261707850036320606710048, −3.28812325925687832005796861536, −2.32434460700083531801700912666, −0.989821728392459052687789830082, 0.60509343885033153658405191336, 1.96783202538834252400209495587, 2.97788022548633321213588898163, 3.75669885852663720638418844321, 4.70184652749487880805929646553, 4.91874578654018494805348891164, 6.01392907159692633807105388869, 6.93302139257622105410762181870, 7.42616976418205099038667147493, 8.016873942583136678315809622254

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