Properties

Label 2-48e2-8.3-c2-0-30
Degree $2$
Conductor $2304$
Sign $0.707 - 0.707i$
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  − 6i·5-s + 6.92i·7-s + 20.7·11-s + 14i·13-s + 6·17-s + 6.92·19-s − 11·25-s + 30i·29-s − 20.7i·31-s + 41.5·35-s + 26i·37-s − 54·41-s − 20.7·43-s + 41.5i·47-s + 1.00·49-s + ⋯
L(s)  = 1  − 1.20i·5-s + 0.989i·7-s + 1.88·11-s + 1.07i·13-s + 0.352·17-s + 0.364·19-s − 0.440·25-s + 1.03i·29-s − 0.670i·31-s + 1.18·35-s + 0.702i·37-s − 1.31·41-s − 0.483·43-s + 0.884i·47-s + 0.0204·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $0.707 - 0.707i$
Analytic conductor: \(62.7794\)
Root analytic conductor: \(7.92334\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{2304} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2304,\ (\ :1),\ 0.707 - 0.707i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.227361072\)
\(L(\frac12)\) \(\approx\) \(2.227361072\)
\(L(2)\) 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 \( 1 + 6iT - 25T^{2} \)
7 \( 1 - 6.92iT - 49T^{2} \)
11 \( 1 - 20.7T + 121T^{2} \)
13 \( 1 - 14iT - 169T^{2} \)
17 \( 1 - 6T + 289T^{2} \)
19 \( 1 - 6.92T + 361T^{2} \)
23 \( 1 - 529T^{2} \)
29 \( 1 - 30iT - 841T^{2} \)
31 \( 1 + 20.7iT - 961T^{2} \)
37 \( 1 - 26iT - 1.36e3T^{2} \)
41 \( 1 + 54T + 1.68e3T^{2} \)
43 \( 1 + 20.7T + 1.84e3T^{2} \)
47 \( 1 - 41.5iT - 2.20e3T^{2} \)
53 \( 1 - 18iT - 2.80e3T^{2} \)
59 \( 1 + 20.7T + 3.48e3T^{2} \)
61 \( 1 - 70iT - 3.72e3T^{2} \)
67 \( 1 + 117.T + 4.48e3T^{2} \)
71 \( 1 - 83.1iT - 5.04e3T^{2} \)
73 \( 1 + 82T + 5.32e3T^{2} \)
79 \( 1 + 76.2iT - 6.24e3T^{2} \)
83 \( 1 - 20.7T + 6.88e3T^{2} \)
89 \( 1 - 114T + 7.92e3T^{2} \)
97 \( 1 - 34T + 9.40e3T^{2} \)
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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

−9.012824043553209784234537466042, −8.512781705020459727766010490498, −7.35549819984374230546526931395, −6.48530220413066196142279280719, −5.80728262334941218987575966273, −4.86134871287296512603279605489, −4.23544502250961265704144209081, −3.21091273383519435551855237727, −1.78471396368394939442634467881, −1.15199731311521595090047502537, 0.60690585996871679307108533713, 1.75315506593520178628234586193, 3.20161174666350340422359855763, 3.59474957724721315819455328579, 4.57687490804278990761073451145, 5.78417763685452092199605828084, 6.57265078280178728174391676861, 7.09071484695040759522801777734, 7.77517741078927428306188947990, 8.728830931614808026578997963606

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