Properties

Label 2-5520-92.91-c1-0-11
Degree $2$
Conductor $5520$
Sign $-0.383 - 0.923i$
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 − 1.58·7-s − 9-s + 2.94·11-s − 1.23·13-s + 15-s + 1.25i·17-s − 5.26·19-s + 1.58i·21-s + (−1.83 − 4.42i)23-s − 25-s + i·27-s + 7.54·29-s − 0.690i·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + 0.447i·5-s − 0.598·7-s − 0.333·9-s + 0.887·11-s − 0.341·13-s + 0.258·15-s + 0.304i·17-s − 1.20·19-s + 0.345i·21-s + (−0.383 − 0.923i)23-s − 0.200·25-s + 0.192i·27-s + 1.40·29-s − 0.123i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.383 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.383 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5520\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.383 - 0.923i$
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: \((2,\ 5520,\ (\ :1/2),\ -0.383 - 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6607073728\)
\(L(\frac12)\) \(\approx\) \(0.6607073728\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.83 + 4.42i)T \)
good7 \( 1 + 1.58T + 7T^{2} \)
11 \( 1 - 2.94T + 11T^{2} \)
13 \( 1 + 1.23T + 13T^{2} \)
17 \( 1 - 1.25iT - 17T^{2} \)
19 \( 1 + 5.26T + 19T^{2} \)
29 \( 1 - 7.54T + 29T^{2} \)
31 \( 1 + 0.690iT - 31T^{2} \)
37 \( 1 + 4.83iT - 37T^{2} \)
41 \( 1 - 6.98T + 41T^{2} \)
43 \( 1 + 4.06T + 43T^{2} \)
47 \( 1 - 8.95iT - 47T^{2} \)
53 \( 1 - 10.8iT - 53T^{2} \)
59 \( 1 - 2.33iT - 59T^{2} \)
61 \( 1 - 5.28iT - 61T^{2} \)
67 \( 1 - 9.49T + 67T^{2} \)
71 \( 1 + 13.5iT - 71T^{2} \)
73 \( 1 + 9.51T + 73T^{2} \)
79 \( 1 - 3.71T + 79T^{2} \)
83 \( 1 + 16.5T + 83T^{2} \)
89 \( 1 - 12.2iT - 89T^{2} \)
97 \( 1 - 7.70iT - 97T^{2} \)
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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

−8.319793256126636598007523951380, −7.62982309312331916117511635744, −6.74691051020629719456007648252, −6.40509761477078489405812766314, −5.82683736449491622359216492216, −4.56393271232328804446963742438, −3.98357195137896901125080509917, −2.91380142300257590455824249728, −2.28553674621264413174642475532, −1.12705609165544231903942901081, 0.18167823485823873528634630326, 1.52364791208375558993319913642, 2.64007469507417351355136017241, 3.55791973598600715399106705249, 4.22967124238305021140082244338, 4.93189015662513007024782985344, 5.74376404536656049379967276613, 6.54799097803356455854090653250, 7.02603754271619203580517568502, 8.265810253140565030240927230213

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