Properties

Label 2-5520-92.91-c1-0-49
Degree $2$
Conductor $5520$
Sign $0.985 + 0.171i$
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 + 1.23·7-s − 9-s + 6.11·11-s − 0.591·13-s − 15-s + 4.95i·17-s + 7.62·19-s − 1.23i·21-s + (−4.72 − 0.821i)23-s − 25-s + i·27-s + 4.66·29-s + 7.50i·31-s + ⋯
L(s)  = 1  − 0.577i·3-s − 0.447i·5-s + 0.466·7-s − 0.333·9-s + 1.84·11-s − 0.164·13-s − 0.258·15-s + 1.20i·17-s + 1.74·19-s − 0.269i·21-s + (−0.985 − 0.171i)23-s − 0.200·25-s + 0.192i·27-s + 0.866·29-s + 1.34i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.171i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 + 0.171i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5520\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.985 + 0.171i$
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: \((2,\ 5520,\ (\ :1/2),\ 0.985 + 0.171i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.479149334\)
\(L(\frac12)\) \(\approx\) \(2.479149334\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.72 + 0.821i)T \)
good7 \( 1 - 1.23T + 7T^{2} \)
11 \( 1 - 6.11T + 11T^{2} \)
13 \( 1 + 0.591T + 13T^{2} \)
17 \( 1 - 4.95iT - 17T^{2} \)
19 \( 1 - 7.62T + 19T^{2} \)
29 \( 1 - 4.66T + 29T^{2} \)
31 \( 1 - 7.50iT - 31T^{2} \)
37 \( 1 + 2.86iT - 37T^{2} \)
41 \( 1 - 4.23T + 41T^{2} \)
43 \( 1 - 4.79T + 43T^{2} \)
47 \( 1 + 5.19iT - 47T^{2} \)
53 \( 1 - 7.65iT - 53T^{2} \)
59 \( 1 - 11.5iT - 59T^{2} \)
61 \( 1 - 11.5iT - 61T^{2} \)
67 \( 1 + 2.90T + 67T^{2} \)
71 \( 1 - 15.7iT - 71T^{2} \)
73 \( 1 - 15.1T + 73T^{2} \)
79 \( 1 + 15.6T + 79T^{2} \)
83 \( 1 - 3.91T + 83T^{2} \)
89 \( 1 - 0.663iT - 89T^{2} \)
97 \( 1 - 11.6iT - 97T^{2} \)
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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

−8.191510119042603372926250877019, −7.38242556186632849473834531763, −6.77058281512154574257583375670, −5.98547629831343174181684128590, −5.40245566300297161515682965077, −4.32678209078180961717929835809, −3.81595424302323159562961606156, −2.71593198035152042629753087195, −1.46934993074984020614483183621, −1.13791351877730029212899168700, 0.77582384956138551952940319761, 1.91430361330510138117738358568, 3.02759718854632342294585005699, 3.69559343354551559020327251555, 4.50189160157132960248731352338, 5.16369160295483069788630232230, 6.10567685788279369457552679632, 6.65979949161419332787667606152, 7.55428642407563570777512629656, 8.057078801083859793283907218927

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