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Properties

Label 2-3456-9.4-c1-0-2
Degree $2$
Conductor $3456$
Sign $-0.999 + 0.0334i$
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  + (−1.34 + 2.32i)5-s + (−2.48 − 4.30i)7-s + (1.26 + 2.19i)11-s + (−2.21 + 3.83i)13-s + 2.43·17-s + 4.18·19-s + (−0.570 + 0.988i)23-s + (−1.09 − 1.89i)25-s + (3.00 + 5.20i)29-s + (2.65 − 4.59i)31-s + 13.3·35-s − 0.241·37-s + (3.21 − 5.56i)41-s + (−5.57 − 9.65i)43-s + (−2.37 − 4.11i)47-s + ⋯
L(s)  = 1  + (−0.599 + 1.03i)5-s + (−0.939 − 1.62i)7-s + (0.382 + 0.662i)11-s + (−0.614 + 1.06i)13-s + 0.590·17-s + 0.959·19-s + (−0.118 + 0.206i)23-s + (−0.218 − 0.378i)25-s + (0.557 + 0.966i)29-s + (0.476 − 0.825i)31-s + 2.25·35-s − 0.0397·37-s + (0.501 − 0.868i)41-s + (−0.849 − 1.47i)43-s + (−0.346 − 0.600i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0334i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0334i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3456\)    =    \(2^{7} \cdot 3^{3}\)
Sign: $-0.999 + 0.0334i$
Analytic conductor: \(27.5962\)
Root analytic conductor: \(5.25321\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3456} (2305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3456,\ (\ :1/2),\ -0.999 + 0.0334i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2077333300\)
\(L(\frac12)\) \(\approx\) \(0.2077333300\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.34 - 2.32i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (2.48 + 4.30i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.26 - 2.19i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.21 - 3.83i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 2.43T + 17T^{2} \)
19 \( 1 - 4.18T + 19T^{2} \)
23 \( 1 + (0.570 - 0.988i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.00 - 5.20i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.65 + 4.59i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 0.241T + 37T^{2} \)
41 \( 1 + (-3.21 + 5.56i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.57 + 9.65i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.37 + 4.11i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 9.38T + 53T^{2} \)
59 \( 1 + (5.40 - 9.36i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.16 + 7.20i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.13 - 1.96i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 4.52T + 71T^{2} \)
73 \( 1 + 3.34T + 73T^{2} \)
79 \( 1 + (-3.14 - 5.44i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.738 - 1.27i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 14.8T + 89T^{2} \)
97 \( 1 + (-5.89 - 10.2i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.206358317360224344685970024722, −7.87987778645012914585079967820, −7.23194533721261184158506967334, −6.98713410360720842889522646467, −6.33213374841964900439702540029, −5.02986039311663461150255594065, −4.07602902449550146906474717682, −3.60656381972858555720677169569, −2.78031156465043554137694943268, −1.36593190528152791893678747467, 0.06842960997739887896908221119, 1.27798781256043520886734150105, 2.93417392143305974817684032438, 3.10840024567238783865153136557, 4.51752808124185457393620331973, 5.21122831389031435772415964982, 5.91260662849047430699545449677, 6.50475880216188513291032099898, 7.84251175120834862760588022373, 8.183384614515432981256693018321

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