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Properties

Label 2-3456-3.2-c0-0-6
Degree $2$
Conductor $3456$
Sign $i$
Analytic cond. $1.72476$
Root an. cond. $1.31330$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41i·5-s + 7-s − 1.41i·11-s − 13-s + 1.41i·17-s + 19-s − 1.41i·23-s − 1.00·25-s − 1.41i·35-s + 37-s − 1.41i·47-s − 2.00·55-s + 1.41i·59-s − 61-s + 1.41i·65-s + ⋯
L(s)  = 1  − 1.41i·5-s + 7-s − 1.41i·11-s − 13-s + 1.41i·17-s + 19-s − 1.41i·23-s − 1.00·25-s − 1.41i·35-s + 37-s − 1.41i·47-s − 2.00·55-s + 1.41i·59-s − 61-s + 1.41i·65-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3456\)    =    \(2^{7} \cdot 3^{3}\)
Sign: $i$
Analytic conductor: \(1.72476\)
Root analytic conductor: \(1.31330\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3456} (1025, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3456,\ (\ :0),\ i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.331435744\)
\(L(\frac12)\) \(\approx\) \(1.331435744\)
\(L(1)\) 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 \( 1 + 1.41iT - T^{2} \)
7 \( 1 - T + T^{2} \)
11 \( 1 + 1.41iT - T^{2} \)
13 \( 1 + T + T^{2} \)
17 \( 1 - 1.41iT - T^{2} \)
19 \( 1 - T + T^{2} \)
23 \( 1 + 1.41iT - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - T + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + 1.41iT - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - 1.41iT - T^{2} \)
61 \( 1 + T + T^{2} \)
67 \( 1 + T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 + T + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - 1.41iT - T^{2} \)
97 \( 1 - T + T^{2} \)
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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

−8.504661437052334617949924525133, −8.127191775113809539677968091454, −7.30279783871197194502179649900, −6.07871793938526732977681027151, −5.52614169383465781048738872582, −4.73058438806257099111888522893, −4.19431923490939779961033756969, −2.99505709305496921544176677316, −1.77977260657371495255769302935, −0.820625848993297335438048596796, 1.63741105153317875961976214680, 2.57215451222796509609759778988, 3.25136025628516738698310533741, 4.55819348057348127226940896151, 4.99084780852044092518870240774, 5.99882362863376641276376157357, 7.04657764784122286639101842200, 7.51593236583452666711276114846, 7.70298054382432092337500119961, 9.208827406886125610736311253592

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