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Properties

Label 2-3456-9.7-c1-0-7
Degree $2$
Conductor $3456$
Sign $0.824 - 0.566i$
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  + (−2.22 − 3.84i)5-s + (1.45 − 2.51i)7-s + (−1.08 + 1.87i)11-s + (1.96 + 3.40i)13-s − 1.79·17-s − 1.76·19-s + (3.44 + 5.96i)23-s + (−7.36 + 12.7i)25-s + (−2.87 + 4.97i)29-s + (3.27 + 5.67i)31-s − 12.9·35-s + 2.51·37-s + (3.68 + 6.38i)41-s + (−2.53 + 4.39i)43-s + (4.98 − 8.63i)47-s + ⋯
L(s)  = 1  + (−0.993 − 1.71i)5-s + (0.549 − 0.952i)7-s + (−0.326 + 0.565i)11-s + (0.544 + 0.943i)13-s − 0.435·17-s − 0.405·19-s + (0.717 + 1.24i)23-s + (−1.47 + 2.54i)25-s + (−0.533 + 0.924i)29-s + (0.588 + 1.01i)31-s − 2.18·35-s + 0.413·37-s + (0.575 + 0.996i)41-s + (−0.386 + 0.669i)43-s + (0.727 − 1.25i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.824 - 0.566i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.824 - 0.566i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3456\)    =    \(2^{7} \cdot 3^{3}\)
Sign: $0.824 - 0.566i$
Analytic conductor: \(27.5962\)
Root analytic conductor: \(5.25321\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3456} (1153, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3456,\ (\ :1/2),\ 0.824 - 0.566i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.082740934\)
\(L(\frac12)\) \(\approx\) \(1.082740934\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.22 + 3.84i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.45 + 2.51i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.08 - 1.87i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.96 - 3.40i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 1.79T + 17T^{2} \)
19 \( 1 + 1.76T + 19T^{2} \)
23 \( 1 + (-3.44 - 5.96i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.87 - 4.97i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.27 - 5.67i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 2.51T + 37T^{2} \)
41 \( 1 + (-3.68 - 6.38i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.53 - 4.39i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.98 + 8.63i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 3.30T + 53T^{2} \)
59 \( 1 + (2.30 + 3.98i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.87 - 3.24i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.36 + 4.08i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 0.907T + 71T^{2} \)
73 \( 1 + 1.87T + 73T^{2} \)
79 \( 1 + (-1.23 + 2.14i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.09 - 1.89i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 5.30T + 89T^{2} \)
97 \( 1 + (-4.45 + 7.71i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.797595933968058132009316959326, −7.81624429143256514709850792289, −7.47834672684470576948447439823, −6.56456329840776549732618654014, −5.30371777888396218229837480106, −4.70535144987671209337123566914, −4.22910928017543884964011533077, −3.43959620674220740616339068638, −1.68508336391776050089633126595, −1.05982438616860946576956151134, 0.37896173648944961791902943566, 2.39580414457560962054156025723, 2.76725058598340132033155320741, 3.74178479753379285146775892127, 4.53531099948370114557753873123, 5.79111615136289096278658043968, 6.19641083874466474506243852174, 7.09754480540722359140825069150, 7.893791673632573376732747047723, 8.272321085564563692421215209507

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