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Properties

Label 2-48e2-8.5-c1-0-12
Degree $2$
Conductor $2304$
Sign $0.707 - 0.707i$
Analytic cond. $18.3975$
Root an. cond. $4.28923$
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  + 4·7-s + 2i·13-s + 8i·19-s + 5·25-s − 4·31-s + 10i·37-s − 8i·43-s + 9·49-s + 14i·61-s − 16i·67-s + 10·73-s − 4·79-s + 8i·91-s + 14·97-s − 20·103-s + ⋯
L(s)  = 1  + 1.51·7-s + 0.554i·13-s + 1.83i·19-s + 25-s − 0.718·31-s + 1.64i·37-s − 1.21i·43-s + 1.28·49-s + 1.79i·61-s − 1.95i·67-s + 1.17·73-s − 0.450·79-s + 0.838i·91-s + 1.42·97-s − 1.97·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $0.707 - 0.707i$
Analytic conductor: \(18.3975\)
Root analytic conductor: \(4.28923\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2304} (1153, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2304,\ (\ :1/2),\ 0.707 - 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.103273157\)
\(L(\frac12)\) \(\approx\) \(2.103273157\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( 1 - 5T^{2} \)
7 \( 1 - 4T + 7T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 8iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 - 10iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 8iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 - 14iT - 61T^{2} \)
67 \( 1 + 16iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 10T + 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 14T + 97T^{2} \)
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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

−8.916005986848948768046008950158, −8.307966103414778660797876641343, −7.69679145129233238818086061353, −6.86213616931562694844577142041, −5.87882499880039947722159218075, −5.08574403899313205399234114451, −4.36763735268295145413584285301, −3.43142782141411512761217949361, −2.07103624696853833837081078594, −1.30854056585834188079647629170, 0.796379215120468710092917622888, 2.01764736536782508280854678189, 2.98141811776130127913181139901, 4.23419154517948233217288474594, 4.96034234069008173628645381104, 5.54068613132645759369844103135, 6.72794048365914793515736509302, 7.43220654048496619312804663254, 8.151351803223064950989127319721, 8.845305595372282706655235078786

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