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Properties

Label 2-3456-9.4-c1-0-16
Degree $2$
Conductor $3456$
Sign $0.972 - 0.231i$
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  + (1.24 − 2.15i)5-s + (−0.909 − 1.57i)7-s + (0.598 + 1.03i)11-s + (−2.83 + 4.90i)13-s + 5.30·17-s + 4.55·19-s + (−2.01 + 3.48i)23-s + (−0.589 − 1.02i)25-s + (3.01 + 5.22i)29-s + (−2.81 + 4.87i)31-s − 4.51·35-s + 5.18·37-s + (−4.57 + 7.92i)41-s + (−3.99 − 6.91i)43-s + (1.39 + 2.41i)47-s + ⋯
L(s)  = 1  + (0.555 − 0.962i)5-s + (−0.343 − 0.595i)7-s + (0.180 + 0.312i)11-s + (−0.785 + 1.36i)13-s + 1.28·17-s + 1.04·19-s + (−0.419 + 0.727i)23-s + (−0.117 − 0.204i)25-s + (0.559 + 0.969i)29-s + (−0.505 + 0.876i)31-s − 0.763·35-s + 0.851·37-s + (−0.714 + 1.23i)41-s + (−0.608 − 1.05i)43-s + (0.203 + 0.352i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 - 0.231i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.972 - 0.231i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3456\)    =    \(2^{7} \cdot 3^{3}\)
Sign: $0.972 - 0.231i$
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: \((2,\ 3456,\ (\ :1/2),\ 0.972 - 0.231i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.973018862\)
\(L(\frac12)\) \(\approx\) \(1.973018862\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.24 + 2.15i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.909 + 1.57i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.598 - 1.03i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.83 - 4.90i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 5.30T + 17T^{2} \)
19 \( 1 - 4.55T + 19T^{2} \)
23 \( 1 + (2.01 - 3.48i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.01 - 5.22i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.81 - 4.87i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 5.18T + 37T^{2} \)
41 \( 1 + (4.57 - 7.92i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.99 + 6.91i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.39 - 2.41i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 1.54T + 53T^{2} \)
59 \( 1 + (1.85 - 3.21i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.01 - 6.95i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.91 + 11.9i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 11.1T + 71T^{2} \)
73 \( 1 - 12.3T + 73T^{2} \)
79 \( 1 + (4.36 + 7.56i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8.89 - 15.4i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 0.455T + 89T^{2} \)
97 \( 1 + (1.01 + 1.76i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.795207091143584263359820029436, −7.78535532591772685701549900847, −7.15444375408251701193698223361, −6.46787486791685197644734290222, −5.34434928973211878790798398279, −4.99516285267326212071144636606, −3.99443581330107635776930146503, −3.14509047154204311934302551443, −1.79545493606596124361020994469, −1.07111708194838349104762745741, 0.69164249922236416990403961770, 2.28953828060103904078834212008, 2.89934249974005654633163895362, 3.60259174017837217998256859232, 4.93813902378140041856915263442, 5.78099358388875909261704915221, 6.09762110049751041687399340867, 7.12613828605436396975892051712, 7.78085810476875367384261408304, 8.450226502939733001213989434403

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