Properties

Label 2-48e2-4.3-c2-0-5
Degree $2$
Conductor $2304$
Sign $-1$
Analytic cond. $62.7794$
Root an. cond. $7.92334$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 16.9i·11-s + 2·17-s + 16.9i·19-s − 25·25-s − 46·41-s − 84.8i·43-s + 49·49-s + 84.8i·59-s − 118. i·67-s − 142·73-s + 50.9i·83-s − 146·89-s + 94·97-s − 118. i·107-s − 98·113-s + ⋯
L(s)  = 1  + 1.54i·11-s + 0.117·17-s + 0.893i·19-s − 25-s − 1.12·41-s − 1.97i·43-s + 0.999·49-s + 1.43i·59-s − 1.77i·67-s − 1.94·73-s + 0.613i·83-s − 1.64·89-s + 0.969·97-s − 1.11i·107-s − 0.867·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $-1$
Analytic conductor: \(62.7794\)
Root analytic conductor: \(7.92334\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{2304} (1279, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2304,\ (\ :1),\ -1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.4471725617\)
\(L(\frac12)\) \(\approx\) \(0.4471725617\)
\(L(2)\) 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 \( 1 + 25T^{2} \)
7 \( 1 - 49T^{2} \)
11 \( 1 - 16.9iT - 121T^{2} \)
13 \( 1 + 169T^{2} \)
17 \( 1 - 2T + 289T^{2} \)
19 \( 1 - 16.9iT - 361T^{2} \)
23 \( 1 - 529T^{2} \)
29 \( 1 + 841T^{2} \)
31 \( 1 - 961T^{2} \)
37 \( 1 + 1.36e3T^{2} \)
41 \( 1 + 46T + 1.68e3T^{2} \)
43 \( 1 + 84.8iT - 1.84e3T^{2} \)
47 \( 1 - 2.20e3T^{2} \)
53 \( 1 + 2.80e3T^{2} \)
59 \( 1 - 84.8iT - 3.48e3T^{2} \)
61 \( 1 + 3.72e3T^{2} \)
67 \( 1 + 118. iT - 4.48e3T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 + 142T + 5.32e3T^{2} \)
79 \( 1 - 6.24e3T^{2} \)
83 \( 1 - 50.9iT - 6.88e3T^{2} \)
89 \( 1 + 146T + 7.92e3T^{2} \)
97 \( 1 - 94T + 9.40e3T^{2} \)
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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

−9.235384836471444207042729171731, −8.448139312718871238150161127569, −7.52720701895520787846921348283, −7.08125241648226899241500919212, −6.05503360488972100009877396828, −5.27411531435400442767236022139, −4.35655484238531751557438516364, −3.62108881631408503189877016112, −2.33839353386164092967432426258, −1.53426115256905361727109725381, 0.10814679148167641565044134418, 1.27201985630527599185413888042, 2.63214251248987863495303488788, 3.41815580981282707536706100955, 4.37285108708539719384653675937, 5.38520497966722607213080345193, 6.05175578867713213553138427863, 6.83616088820026541030727861164, 7.78625039234257766624864220524, 8.454812691377401612463180879601

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