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Properties

Label 2-2184-13.3-c1-0-19
Degree $2$
Conductor $2184$
Sign $0.968 + 0.249i$
Analytic cond. $17.4393$
Root an. cond. $4.17604$
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  + (0.5 + 0.866i)3-s − 2.46·5-s + (−0.5 + 0.866i)7-s + (−0.499 + 0.866i)9-s + (0.0102 + 0.0178i)11-s + (−0.0102 − 3.60i)13-s + (−1.23 − 2.13i)15-s + (−0.198 + 0.343i)17-s + (2.96 − 5.14i)19-s − 0.999·21-s + (1.93 + 3.34i)23-s + 1.09·25-s − 0.999·27-s + (3.01 + 5.22i)29-s − 4.22·31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s − 1.10·5-s + (−0.188 + 0.327i)7-s + (−0.166 + 0.288i)9-s + (0.00310 + 0.00537i)11-s + (−0.00285 − 0.999i)13-s + (−0.318 − 0.551i)15-s + (−0.0481 + 0.0833i)17-s + (0.680 − 1.17i)19-s − 0.218·21-s + (0.402 + 0.697i)23-s + 0.218·25-s − 0.192·27-s + (0.559 + 0.969i)29-s − 0.759·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.249i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 + 0.249i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2184\)    =    \(2^{3} \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.968 + 0.249i$
Analytic conductor: \(17.4393\)
Root analytic conductor: \(4.17604\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2184} (1849, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2184,\ (\ :1/2),\ 0.968 + 0.249i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.322797199\)
\(L(\frac12)\) \(\approx\) \(1.322797199\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (0.0102 + 3.60i)T \)
good5 \( 1 + 2.46T + 5T^{2} \)
11 \( 1 + (-0.0102 - 0.0178i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.198 - 0.343i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.96 + 5.14i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.93 - 3.34i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.01 - 5.22i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4.22T + 31T^{2} \)
37 \( 1 + (5.32 + 9.22i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.62 + 4.53i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.15 + 2.00i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 10.1T + 47T^{2} \)
53 \( 1 - 13.0T + 53T^{2} \)
59 \( 1 + (-6.71 + 11.6i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.06 - 10.4i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.79 - 8.29i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.79 + 3.10i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 14.7T + 73T^{2} \)
79 \( 1 - 3.50T + 79T^{2} \)
83 \( 1 - 12.0T + 83T^{2} \)
89 \( 1 + (-5.81 - 10.0i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.27 + 9.13i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.936237936069923053077297615218, −8.396185772900368906467366484465, −7.41168687261309418455873360899, −7.04372402563511330000525639313, −5.55979160391433366063097789606, −5.12772224622938573708458273230, −3.91249660031372705466796311145, −3.40229069139838854185428193711, −2.40202312900170539245783813447, −0.59179673339151326998196183391, 0.921797690659536711726384550024, 2.22487581021553732493321013425, 3.43883445323151380556519753952, 4.02651709218368600195388357005, 4.97802629780010129043031968944, 6.18692500397797347994128814655, 6.88784516414581120164094939806, 7.60722707604105847960012756128, 8.184440601660348276453588792322, 8.936587231655859086859498993252

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