Properties

Label 2-48e2-8.3-c2-0-4
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  + 8i·5-s + 10i·13-s − 16·17-s − 39·25-s + 40i·29-s − 70i·37-s − 80·41-s + 49·49-s + 56i·53-s + 22i·61-s − 80·65-s − 110·73-s − 128i·85-s + 160·89-s − 130·97-s + ⋯
L(s)  = 1  + 1.60i·5-s + 0.769i·13-s − 0.941·17-s − 1.56·25-s + 1.37i·29-s − 1.89i·37-s − 1.95·41-s + 0.999·49-s + 1.05i·53-s + 0.360i·61-s − 1.23·65-s − 1.50·73-s − 1.50i·85-s + 1.79·89-s − 1.34·97-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

\(L(\frac{3}{2})\) \(\approx\) \(0.4287745754\)
\(L(\frac12)\) \(\approx\) \(0.4287745754\)
\(L(2)\) 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 \( 1 - 8iT - 25T^{2} \)
7 \( 1 - 49T^{2} \)
11 \( 1 + 121T^{2} \)
13 \( 1 - 10iT - 169T^{2} \)
17 \( 1 + 16T + 289T^{2} \)
19 \( 1 + 361T^{2} \)
23 \( 1 - 529T^{2} \)
29 \( 1 - 40iT - 841T^{2} \)
31 \( 1 - 961T^{2} \)
37 \( 1 + 70iT - 1.36e3T^{2} \)
41 \( 1 + 80T + 1.68e3T^{2} \)
43 \( 1 + 1.84e3T^{2} \)
47 \( 1 - 2.20e3T^{2} \)
53 \( 1 - 56iT - 2.80e3T^{2} \)
59 \( 1 + 3.48e3T^{2} \)
61 \( 1 - 22iT - 3.72e3T^{2} \)
67 \( 1 + 4.48e3T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 + 110T + 5.32e3T^{2} \)
79 \( 1 - 6.24e3T^{2} \)
83 \( 1 + 6.88e3T^{2} \)
89 \( 1 - 160T + 7.92e3T^{2} \)
97 \( 1 + 130T + 9.40e3T^{2} \)
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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

−9.232209136851508745226665343860, −8.663925471996682732216225294959, −7.46170853131590429054819953335, −7.00292305791717329107484390345, −6.40632977768699523168355530942, −5.50588378388715542281210878335, −4.36135707365332285886039434454, −3.52158219445314296362992092321, −2.65013416968999020829653281068, −1.77237310731648065597035701029, 0.10680510751916947564980557105, 1.12192597085330551879615934889, 2.20624476668426273428652406136, 3.48727403363898510937007883629, 4.52149573289662922536254066216, 5.01722402598296722380591871320, 5.87291774803100614173443823406, 6.73710319460723332686557801606, 7.85270983954925281257458878700, 8.410717922423940163256048948958

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