Properties

Label 2-18e2-36.31-c2-0-44
Degree $2$
Conductor $324$
Sign $-0.0211 + 0.999i$
Analytic cond. $8.82836$
Root an. cond. $2.97125$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.82 − 0.826i)2-s + (2.63 − 3.00i)4-s + (3.61 − 6.26i)5-s + (5.95 − 3.44i)7-s + (2.31 − 7.65i)8-s + (1.41 − 14.4i)10-s + (−15.5 + 8.97i)11-s + (−3.83 + 6.64i)13-s + (8.01 − 11.1i)14-s + (−2.11 − 15.8i)16-s + 1.43·17-s + 13.8i·19-s + (−9.32 − 27.4i)20-s + (−20.9 + 29.1i)22-s + (14.3 + 8.29i)23-s + ⋯
L(s)  = 1  + (0.910 − 0.413i)2-s + (0.658 − 0.752i)4-s + (0.723 − 1.25i)5-s + (0.851 − 0.491i)7-s + (0.289 − 0.957i)8-s + (0.141 − 1.44i)10-s + (−1.41 + 0.816i)11-s + (−0.295 + 0.511i)13-s + (0.572 − 0.799i)14-s + (−0.131 − 0.991i)16-s + 0.0845·17-s + 0.731i·19-s + (−0.466 − 1.37i)20-s + (−0.950 + 1.32i)22-s + (0.624 + 0.360i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0211 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0211 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $-0.0211 + 0.999i$
Analytic conductor: \(8.82836\)
Root analytic conductor: \(2.97125\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :1),\ -0.0211 + 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.23874 - 2.28655i\)
\(L(\frac12)\) \(\approx\) \(2.23874 - 2.28655i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.82 + 0.826i)T \)
3 \( 1 \)
good5 \( 1 + (-3.61 + 6.26i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 + (-5.95 + 3.44i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (15.5 - 8.97i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (3.83 - 6.64i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 - 1.43T + 289T^{2} \)
19 \( 1 - 13.8iT - 361T^{2} \)
23 \( 1 + (-14.3 - 8.29i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (18.7 + 32.4i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (-46.7 - 26.9i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 - 44.4T + 1.36e3T^{2} \)
41 \( 1 + (-28.4 + 49.3i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (58.5 - 33.8i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-8.17 + 4.71i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 7.82T + 2.80e3T^{2} \)
59 \( 1 + (-19.4 - 11.2i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-33.5 - 58.1i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-59.9 - 34.6i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 7.14iT - 5.04e3T^{2} \)
73 \( 1 + 80.4T + 5.32e3T^{2} \)
79 \( 1 + (-40.5 + 23.4i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (123. - 71.0i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 42.1T + 7.92e3T^{2} \)
97 \( 1 + (31.4 + 54.4i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−11.33929282406327249049576790199, −10.22717760237739447308693413382, −9.649868573088014514434029497888, −8.253646808413345216455225544056, −7.25872882299148412251941226049, −5.77524859987531270745432038134, −4.96418147185366715965580215122, −4.32947475026475249773336268968, −2.39840201765888921482566479590, −1.26801998830538892209240012513, 2.42694053300384852348994950769, 3.04472548610937927951439817680, 4.88826636551815172479780668923, 5.64941150754795982368567735196, 6.59923678355478474163004878112, 7.66377554843521695513998770497, 8.456706833653557464246778799992, 10.05381856625299143598376667705, 11.00343587344759360940574854098, 11.43114906174000633803061390411

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