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Properties

Label 2-3456-8.5-c1-0-14
Degree $2$
Conductor $3456$
Sign $0.707 - 0.707i$
Analytic cond. $27.5962$
Root an. cond. $5.25321$
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  − 3i·5-s − 7-s + 5i·11-s − 4i·13-s + 4·17-s + 8i·19-s − 8·23-s − 4·25-s + 2i·29-s + 7·31-s + 3i·35-s + 8i·37-s − 4i·43-s − 4·47-s − 6·49-s + ⋯
L(s)  = 1  − 1.34i·5-s − 0.377·7-s + 1.50i·11-s − 1.10i·13-s + 0.970·17-s + 1.83i·19-s − 1.66·23-s − 0.800·25-s + 0.371i·29-s + 1.25·31-s + 0.507i·35-s + 1.31i·37-s − 0.609i·43-s − 0.583·47-s − 0.857·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3456\)    =    \(2^{7} \cdot 3^{3}\)
Sign: $0.707 - 0.707i$
Analytic conductor: \(27.5962\)
Root analytic conductor: \(5.25321\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3456} (1729, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3456,\ (\ :1/2),\ 0.707 - 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.373597873\)
\(L(\frac12)\) \(\approx\) \(1.373597873\)
\(L(\frac{3}{2})\) 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 \( 1 + 3iT - 5T^{2} \)
7 \( 1 + T + 7T^{2} \)
11 \( 1 - 5iT - 11T^{2} \)
13 \( 1 + 4iT - 13T^{2} \)
17 \( 1 - 4T + 17T^{2} \)
19 \( 1 - 8iT - 19T^{2} \)
23 \( 1 + 8T + 23T^{2} \)
29 \( 1 - 2iT - 29T^{2} \)
31 \( 1 - 7T + 31T^{2} \)
37 \( 1 - 8iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 + 4T + 47T^{2} \)
53 \( 1 - 7iT - 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 12iT - 67T^{2} \)
71 \( 1 + 4T + 71T^{2} \)
73 \( 1 - 15T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 + iT - 83T^{2} \)
89 \( 1 + 12T + 89T^{2} \)
97 \( 1 - 7T + 97T^{2} \)
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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

−8.403016936834538288469810724710, −8.121010662458316823886881077367, −7.41690682663416185895850672719, −6.27470084574632242375286561871, −5.61693557461496464021712863491, −4.87725056324764156445244084425, −4.14748623122166565950796400327, −3.24585859912924708263753260273, −1.93406576309082238562826725070, −1.06985127984863526346976272214, 0.46305241842986166583116479799, 2.11582966785992878882918906320, 3.00373317061764681996532197417, 3.57179595346358459896554092565, 4.57348364286840330302588909792, 5.73022778056249847735557680364, 6.40831491063383685968229180649, 6.79750842604430871873806447742, 7.75328421705054306003732659572, 8.409005127505659549667901572755

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