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Properties

Label 2-3456-9.7-c1-0-26
Degree $2$
Conductor $3456$
Sign $-0.447 + 0.894i$
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  + (−0.268 − 0.464i)5-s + (−2.35 + 4.07i)7-s + (−2.59 + 4.50i)11-s + (−0.778 − 1.34i)13-s − 0.695·17-s − 5.80·19-s + (4.42 + 7.66i)23-s + (2.35 − 4.08i)25-s + (−1.92 + 3.32i)29-s + (−2.77 − 4.79i)31-s + 2.52·35-s + 4.09·37-s + (−1.01 − 1.74i)41-s + (3.71 − 6.43i)43-s + (−0.186 + 0.322i)47-s + ⋯
L(s)  = 1  + (−0.119 − 0.207i)5-s + (−0.888 + 1.53i)7-s + (−0.783 + 1.35i)11-s + (−0.215 − 0.373i)13-s − 0.168·17-s − 1.33·19-s + (0.923 + 1.59i)23-s + (0.471 − 0.816i)25-s + (−0.356 + 0.618i)29-s + (−0.497 − 0.861i)31-s + 0.426·35-s + 0.672·37-s + (−0.157 − 0.273i)41-s + (0.566 − 0.981i)43-s + (−0.0271 + 0.0470i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3456\)    =    \(2^{7} \cdot 3^{3}\)
Sign: $-0.447 + 0.894i$
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: \((2,\ 3456,\ (\ :1/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.05528590298\)
\(L(\frac12)\) \(\approx\) \(0.05528590298\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.268 + 0.464i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (2.35 - 4.07i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.59 - 4.50i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.778 + 1.34i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 0.695T + 17T^{2} \)
19 \( 1 + 5.80T + 19T^{2} \)
23 \( 1 + (-4.42 - 7.66i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.92 - 3.32i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.77 + 4.79i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 4.09T + 37T^{2} \)
41 \( 1 + (1.01 + 1.74i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.71 + 6.43i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.186 - 0.322i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 5.30T + 53T^{2} \)
59 \( 1 + (2.57 + 4.45i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.921 - 1.59i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.79 + 10.0i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.6T + 71T^{2} \)
73 \( 1 - 4.40T + 73T^{2} \)
79 \( 1 + (3.32 - 5.75i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.28 + 9.14i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 7.30T + 89T^{2} \)
97 \( 1 + (7.81 - 13.5i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.400181039665761943641949475471, −7.63056140502600181166707048179, −6.85280149253910454665836910751, −6.03954872632066041315875863860, −5.31874197874815106181444888331, −4.67733272484153155383476706364, −3.53066108159965576968540429179, −2.56536225194077089784404762677, −1.97212053000405840847610399628, −0.01895351186931013346941436792, 0.966880929232271388626880932643, 2.59693818872052013758856371058, 3.31626942695174918821495140299, 4.14679666127297851222473672417, 4.88247518863452779822296123894, 6.08628144509697208791176458470, 6.60647591868859355401780835297, 7.26499598756061733671134836675, 8.062902290569917606141294155061, 8.788797193659924633482502945384

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