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Properties

Label 2-48e2-24.11-c1-0-23
Degree $2$
Conductor $2304$
Sign $0.169 + 0.985i$
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  + 1.41·5-s − 4i·7-s + 5.65i·11-s − 4i·13-s − 4.24i·17-s + 5.65·23-s − 2.99·25-s + 1.41·29-s + 4i·31-s − 5.65i·35-s − 6i·37-s − 9.89i·41-s + 8·43-s − 5.65·47-s − 9·49-s + ⋯
L(s)  = 1  + 0.632·5-s − 1.51i·7-s + 1.70i·11-s − 1.10i·13-s − 1.02i·17-s + 1.17·23-s − 0.599·25-s + 0.262·29-s + 0.718i·31-s − 0.956i·35-s − 0.986i·37-s − 1.54i·41-s + 1.21·43-s − 0.825·47-s − 1.28·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.169 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.169 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $0.169 + 0.985i$
Analytic conductor: \(18.3975\)
Root analytic conductor: \(4.28923\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2304} (1151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2304,\ (\ :1/2),\ 0.169 + 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.826786966\)
\(L(\frac12)\) \(\approx\) \(1.826786966\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.41T + 5T^{2} \)
7 \( 1 + 4iT - 7T^{2} \)
11 \( 1 - 5.65iT - 11T^{2} \)
13 \( 1 + 4iT - 13T^{2} \)
17 \( 1 + 4.24iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 5.65T + 23T^{2} \)
29 \( 1 - 1.41T + 29T^{2} \)
31 \( 1 - 4iT - 31T^{2} \)
37 \( 1 + 6iT - 37T^{2} \)
41 \( 1 + 9.89iT - 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + 5.65T + 47T^{2} \)
53 \( 1 - 4.24T + 53T^{2} \)
59 \( 1 + 11.3iT - 59T^{2} \)
61 \( 1 - 2iT - 61T^{2} \)
67 \( 1 + 8T + 67T^{2} \)
71 \( 1 + 5.65T + 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 + 4iT - 79T^{2} \)
83 \( 1 - 5.65iT - 83T^{2} \)
89 \( 1 + 4.24iT - 89T^{2} \)
97 \( 1 + 8T + 97T^{2} \)
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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

−9.037688831992506912855931688971, −7.77118793662826549136049079778, −7.27593242946436495477494380787, −6.77988488333197482749477774426, −5.55193137489351761127452833223, −4.84148087304427675768938757511, −4.05013072201602587744553046371, −2.97350149005560433578173383258, −1.85397237138863869470768367494, −0.64612685003027003406486146624, 1.37224929802246596944659506111, 2.44761828909052973057732250716, 3.23054301071978043842013761903, 4.41977581915713738266791530417, 5.48849172960017969330937475123, 6.04572941218993921300572130834, 6.49482892632480816257068308283, 7.83326547206627435374334537266, 8.680192195260568064563270034163, 8.974905994026304512665107849248

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