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Properties

Label 2-3456-9.4-c1-0-47
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: $\chi_{3456} (2305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3456,\ (\ :1/2),\ -0.824 - 0.566i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6208944026\)
\(L(\frac12)\) \(\approx\) \(0.6208944026\)
\(L(\frac{3}{2})\) 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

−8.460608858054698435254660752164, −7.32216486723398486713847024643, −6.67547750348821360863504174772, −5.71379798273385891592644244784, −5.16082866326247063669329756087, −4.35463068991213853628413042362, −3.59046531329492355190331172909, −2.14964487192024144734303529243, −1.35697300458293840495975827863, −0.16998066864903456882720050655, 2.10436759825135598031628980143, 2.55197282563960301787295334088, 3.18312277687553584191026367259, 4.46058811499914596574486244445, 5.53162049957203901288495483299, 6.22892505104407661510169791292, 6.51642406862345598308337084970, 7.51478088294929160534361195249, 8.150343406060073826269744038570, 9.313684133142525308865993597043

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