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Properties

Label 2-5520-92.91-c1-0-28
Degree $2$
Conductor $5520$
Sign $0.399 - 0.916i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.399 - 0.916i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.399 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5520\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.399 - 0.916i$
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: \((2,\ 5520,\ (\ :1/2),\ 0.399 - 0.916i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.070839742\)
\(L(\frac12)\) \(\approx\) \(1.070839742\)
\(L(\frac{3}{2})\) 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

−8.375700172610754343584741269593, −7.55924571904113809974603394968, −6.73896932940025509055946587422, −6.23002308432979771631760783777, −5.17173278453928669589850532614, −4.83690212782235742172608861124, −3.55674847011634575485160445613, −3.31792933505568317034005423216, −2.17968682486758955809339032441, −0.74088826228804624397200708355, 0.39125992195438055677969713629, 1.77409825943784783741617551706, 2.63824479501574833084258709684, 3.52810297550396751538070575731, 4.05381042874993973059502744155, 5.39272705775613297366244476785, 6.04467486518770988174102457089, 6.65071531945293373150023986725, 7.13733346961646591650124012381, 7.904808099441652479441804683399

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