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Properties

Label 2-3456-48.5-c0-0-3
Degree $2$
Conductor $3456$
Sign $0.382 + 0.923i$
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  + (0.707 − 0.707i)5-s − i·7-s + (0.707 − 0.707i)11-s + (1 − i)13-s + 1.41i·17-s − 31-s + (−0.707 − 0.707i)35-s + (−1 − i)43-s + 1.41i·47-s + (−0.707 + 0.707i)53-s − 1.00i·55-s − 1.41i·65-s + (1 − i)67-s − 1.41·71-s − i·73-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)5-s − i·7-s + (0.707 − 0.707i)11-s + (1 − i)13-s + 1.41i·17-s − 31-s + (−0.707 − 0.707i)35-s + (−1 − i)43-s + 1.41i·47-s + (−0.707 + 0.707i)53-s − 1.00i·55-s − 1.41i·65-s + (1 − i)67-s − 1.41·71-s − i·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3456\)    =    \(2^{7} \cdot 3^{3}\)
Sign: $0.382 + 0.923i$
Analytic conductor: \(1.72476\)
Root analytic conductor: \(1.31330\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3456} (1889, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3456,\ (\ :0),\ 0.382 + 0.923i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.549753591\)
\(L(\frac12)\) \(\approx\) \(1.549753591\)
\(L(1)\) 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 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
7 \( 1 + iT - T^{2} \)
11 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
13 \( 1 + (-1 + i)T - iT^{2} \)
17 \( 1 - 1.41iT - T^{2} \)
19 \( 1 - iT^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + iT^{2} \)
31 \( 1 + T + T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (1 + i)T + iT^{2} \)
47 \( 1 - 1.41iT - T^{2} \)
53 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
59 \( 1 - iT^{2} \)
61 \( 1 - iT^{2} \)
67 \( 1 + (-1 + i)T - iT^{2} \)
71 \( 1 + 1.41T + T^{2} \)
73 \( 1 + iT - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
89 \( 1 + 1.41T + T^{2} \)
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

−8.628274800997193263979665910746, −8.040655721671798578783149458312, −7.16317801986797350076206849428, −6.11411862006949073534920299243, −5.85472688345247317678854119381, −4.82831800522307936409262953472, −3.83758255451358024999713724669, −3.34579123097174486851651422760, −1.73380066805752416871519918833, −1.01253875435238565152241279947, 1.63504707328161494669329486038, 2.35703274705881223463641250140, 3.29158392515853272049768848592, 4.30517895314614980040945814815, 5.23697828147467300500302241865, 5.99032145995315061026829096978, 6.73054018483602537868310888928, 7.12166866925790237804724678281, 8.346989752036580390146453519136, 9.029666057074493365817450513900

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