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Properties

Label 2-3456-9.4-c1-0-13
Degree $2$
Conductor $3456$
Sign $0.408 - 0.912i$
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  + (1.07 − 1.86i)5-s + (0.153 + 0.265i)7-s + (2.50 + 4.34i)11-s + (−0.470 + 0.815i)13-s + 4.70·17-s − 1.61·19-s + (−4.08 + 7.06i)23-s + (0.191 + 0.330i)25-s + (2.39 + 4.14i)29-s + (−1.29 + 2.24i)31-s + 0.658·35-s − 10.2·37-s + (−3.86 + 6.69i)41-s + (−0.138 − 0.239i)43-s + (−1.92 − 3.32i)47-s + ⋯
L(s)  = 1  + (0.480 − 0.832i)5-s + (0.0578 + 0.100i)7-s + (0.755 + 1.30i)11-s + (−0.130 + 0.226i)13-s + 1.14·17-s − 0.371·19-s + (−0.851 + 1.47i)23-s + (0.0382 + 0.0661i)25-s + (0.444 + 0.770i)29-s + (−0.233 + 0.403i)31-s + 0.111·35-s − 1.67·37-s + (−0.603 + 1.04i)41-s + (−0.0210 − 0.0364i)43-s + (−0.280 − 0.485i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.408 - 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.408 - 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3456\)    =    \(2^{7} \cdot 3^{3}\)
Sign: $0.408 - 0.912i$
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: \((2,\ 3456,\ (\ :1/2),\ 0.408 - 0.912i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.838681657\)
\(L(\frac12)\) \(\approx\) \(1.838681657\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.07 + 1.86i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.153 - 0.265i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.50 - 4.34i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.470 - 0.815i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 4.70T + 17T^{2} \)
19 \( 1 + 1.61T + 19T^{2} \)
23 \( 1 + (4.08 - 7.06i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.39 - 4.14i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.29 - 2.24i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 10.2T + 37T^{2} \)
41 \( 1 + (3.86 - 6.69i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.138 + 0.239i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.92 + 3.32i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 2.23T + 53T^{2} \)
59 \( 1 + (4.95 - 8.58i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.36 + 9.29i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.02 - 3.50i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.59T + 71T^{2} \)
73 \( 1 + 5.43T + 73T^{2} \)
79 \( 1 + (-8.30 - 14.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.91 - 5.05i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 1.94T + 89T^{2} \)
97 \( 1 + (-7.07 - 12.2i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.823894931733015752452779572722, −8.036901439437140166133492166247, −7.19284002071946159961327533485, −6.58384062700528130134669354296, −5.46451745876436452949173930226, −5.09799159475126723763757113595, −4.13377913611723758433846797520, −3.29577078404258302493780152573, −1.85018457913673630097699410128, −1.38457465701167947577875702318, 0.55261590376457716693097889875, 1.90544465042782234924778558574, 2.92673883834884792203427540680, 3.60106644409565605414040878290, 4.56160276748826343689111004098, 5.72147037936267280433528210062, 6.17415070621085100585890453736, 6.81930528347483514098811989260, 7.76125701628094319390903925236, 8.457635540937281765332742561752

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