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Properties

Label 2-5472-12.11-c1-0-13
Degree $2$
Conductor $5472$
Sign $-0.985 - 0.169i$
Analytic cond. $43.6941$
Root an. cond. $6.61015$
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  + 3.62i·5-s + 0.712i·7-s + 3.62·11-s − 3.02·13-s + 5.59i·17-s − i·19-s + 2.37·23-s − 8.14·25-s + 1.44i·29-s + 1.02i·31-s − 2.58·35-s − 3.02·37-s + 2.01i·41-s + 7.78i·43-s − 6.97·47-s + ⋯
L(s)  = 1  + 1.62i·5-s + 0.269i·7-s + 1.09·11-s − 0.838·13-s + 1.35i·17-s − 0.229i·19-s + 0.494·23-s − 1.62·25-s + 0.269i·29-s + 0.184i·31-s − 0.436·35-s − 0.497·37-s + 0.314i·41-s + 1.18i·43-s − 1.01·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5472 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5472\)    =    \(2^{5} \cdot 3^{2} \cdot 19\)
Sign: $-0.985 - 0.169i$
Analytic conductor: \(43.6941\)
Root analytic conductor: \(6.61015\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5472} (2015, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5472,\ (\ :1/2),\ -0.985 - 0.169i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.378382658\)
\(L(\frac12)\) \(\approx\) \(1.378382658\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19 \( 1 + iT \)
good5 \( 1 - 3.62iT - 5T^{2} \)
7 \( 1 - 0.712iT - 7T^{2} \)
11 \( 1 - 3.62T + 11T^{2} \)
13 \( 1 + 3.02T + 13T^{2} \)
17 \( 1 - 5.59iT - 17T^{2} \)
23 \( 1 - 2.37T + 23T^{2} \)
29 \( 1 - 1.44iT - 29T^{2} \)
31 \( 1 - 1.02iT - 31T^{2} \)
37 \( 1 + 3.02T + 37T^{2} \)
41 \( 1 - 2.01iT - 41T^{2} \)
43 \( 1 - 7.78iT - 43T^{2} \)
47 \( 1 + 6.97T + 47T^{2} \)
53 \( 1 + 2.40iT - 53T^{2} \)
59 \( 1 - 5.80T + 59T^{2} \)
61 \( 1 + 11.8T + 61T^{2} \)
67 \( 1 - 5.42iT - 67T^{2} \)
71 \( 1 + 1.87T + 71T^{2} \)
73 \( 1 - 5.49T + 73T^{2} \)
79 \( 1 - 7.07iT - 79T^{2} \)
83 \( 1 + 5.20T + 83T^{2} \)
89 \( 1 + 2.68iT - 89T^{2} \)
97 \( 1 - 12.9T + 97T^{2} \)
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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

−8.475670327260018290657309861724, −7.62568889185941404492532894487, −6.97090326741199005832470931052, −6.45845187732756547406844047936, −5.87965359250983443731087714022, −4.81971289721668585395017394870, −3.89178258452401357394299674566, −3.21742880183010961216810680406, −2.44618063360398784136233647473, −1.49433871293300972041524742012, 0.37344898734763125636612561571, 1.22415884528600108742922121207, 2.21574021438263846990918801338, 3.44000971845736576625112785225, 4.29855349827477136790976879269, 4.89779814711489717014719195828, 5.43273683669805964951636492797, 6.40607160452122964914997604886, 7.23767874037490481201546784540, 7.78364566976832718256668838237

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