Properties

Label 2-5520-92.91-c1-0-21
Degree $2$
Conductor $5520$
Sign $0.988 - 0.149i$
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 − 1.52·7-s − 9-s − 6.43·11-s − 4.50·13-s − 15-s − 1.65i·17-s + 5.57·19-s + 1.52i·21-s + (2.99 + 3.74i)23-s − 25-s + i·27-s − 2.30·29-s + 1.77i·31-s + ⋯
L(s)  = 1  − 0.577i·3-s − 0.447i·5-s − 0.575·7-s − 0.333·9-s − 1.93·11-s − 1.24·13-s − 0.258·15-s − 0.400i·17-s + 1.27·19-s + 0.332i·21-s + (0.623 + 0.781i)23-s − 0.200·25-s + 0.192i·27-s − 0.427·29-s + 0.318i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.149i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 - 0.149i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5520\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.988 - 0.149i$
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: \((2,\ 5520,\ (\ :1/2),\ 0.988 - 0.149i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8283836053\)
\(L(\frac12)\) \(\approx\) \(0.8283836053\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.99 - 3.74i)T \)
good7 \( 1 + 1.52T + 7T^{2} \)
11 \( 1 + 6.43T + 11T^{2} \)
13 \( 1 + 4.50T + 13T^{2} \)
17 \( 1 + 1.65iT - 17T^{2} \)
19 \( 1 - 5.57T + 19T^{2} \)
29 \( 1 + 2.30T + 29T^{2} \)
31 \( 1 - 1.77iT - 31T^{2} \)
37 \( 1 + 5.43iT - 37T^{2} \)
41 \( 1 + 2.07T + 41T^{2} \)
43 \( 1 + 4.75T + 43T^{2} \)
47 \( 1 - 2.99iT - 47T^{2} \)
53 \( 1 - 2.72iT - 53T^{2} \)
59 \( 1 - 3.99iT - 59T^{2} \)
61 \( 1 + 9.74iT - 61T^{2} \)
67 \( 1 - 3.71T + 67T^{2} \)
71 \( 1 - 2.39iT - 71T^{2} \)
73 \( 1 + 9.23T + 73T^{2} \)
79 \( 1 - 11.4T + 79T^{2} \)
83 \( 1 - 9.35T + 83T^{2} \)
89 \( 1 - 2.66iT - 89T^{2} \)
97 \( 1 - 8.66iT - 97T^{2} \)
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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

−7.83999069499635780566386654148, −7.58633203716671064877054198186, −6.96607297185526196679611052377, −5.92255379046240202924752806897, −5.11286157644548658588256040056, −4.96302531132379982862354418657, −3.42745296662001879080595688140, −2.81604115807354758628022974193, −1.98041879872206863224228077622, −0.63734597319179879331817176826, 0.33009427053751723753196412835, 2.16112533332015219185335078869, 2.93474474291758621104186947731, 3.40982451638614877794772338182, 4.71474598814175792252623279113, 5.10790424669586214715586144624, 5.86599981751370124713323534298, 6.78661684993206069682001588088, 7.49278685519546524123851127491, 8.017385434215877575072291678988

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