Properties

Label 2-5520-92.91-c1-0-94
Degree $2$
Conductor $5520$
Sign $-0.993 + 0.112i$
Analytic cond. $44.0774$
Root an. cond. $6.63908$
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·3-s i·5-s + 3.67·7-s − 9-s − 1.00·11-s − 0.593·13-s − 15-s − 5.94i·17-s + 0.286·19-s − 3.67i·21-s + (−2.84 − 3.85i)23-s − 25-s + i·27-s − 6.36·29-s + 1.69i·31-s + ⋯
L(s)  = 1  − 0.577i·3-s − 0.447i·5-s + 1.38·7-s − 0.333·9-s − 0.303·11-s − 0.164·13-s − 0.258·15-s − 1.44i·17-s + 0.0656·19-s − 0.801i·21-s + (−0.594 − 0.804i)23-s − 0.200·25-s + 0.192i·27-s − 1.18·29-s + 0.304i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.112i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 + 0.112i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5520\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.993 + 0.112i$
Analytic conductor: \(44.0774\)
Root analytic conductor: \(6.63908\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5520} (1471, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5520,\ (\ :1/2),\ -0.993 + 0.112i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.142649579\)
\(L(\frac12)\) \(\approx\) \(1.142649579\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + iT \)
5 \( 1 + iT \)
23 \( 1 + (2.84 + 3.85i)T \)
good7 \( 1 - 3.67T + 7T^{2} \)
11 \( 1 + 1.00T + 11T^{2} \)
13 \( 1 + 0.593T + 13T^{2} \)
17 \( 1 + 5.94iT - 17T^{2} \)
19 \( 1 - 0.286T + 19T^{2} \)
29 \( 1 + 6.36T + 29T^{2} \)
31 \( 1 - 1.69iT - 31T^{2} \)
37 \( 1 - 10.8iT - 37T^{2} \)
41 \( 1 + 9.08T + 41T^{2} \)
43 \( 1 + 2.53T + 43T^{2} \)
47 \( 1 + 9.21iT - 47T^{2} \)
53 \( 1 + 9.48iT - 53T^{2} \)
59 \( 1 - 7.46iT - 59T^{2} \)
61 \( 1 + 10.2iT - 61T^{2} \)
67 \( 1 + 6.43T + 67T^{2} \)
71 \( 1 - 2.38iT - 71T^{2} \)
73 \( 1 - 10.8T + 73T^{2} \)
79 \( 1 + 13.4T + 79T^{2} \)
83 \( 1 + 1.86T + 83T^{2} \)
89 \( 1 - 14.2iT - 89T^{2} \)
97 \( 1 + 0.822iT - 97T^{2} \)
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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

−8.008970597666342619583197947368, −7.12464880205565264086088932682, −6.56325208415627908387077243392, −5.28603358164436061731721650051, −5.16171814615472079432025611138, −4.28647843142289791977082421830, −3.15358281999313859322853921448, −2.15434029380194648899597632849, −1.46077736443625581858354727832, −0.27685361023716269368824665081, 1.54072543656432653970465889706, 2.22480891413156539151446018083, 3.42468694650585058675860847060, 4.08048948527452758892363256034, 4.81970971789210872591159287040, 5.65064912656383571801711873944, 6.09678761878965525438973996991, 7.38408477422978175362402725604, 7.70903708080605650697741759567, 8.485216393735199103592588864424

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