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Properties

Label 2-132-33.2-c1-0-2
Degree $2$
Conductor $132$
Sign $0.980 - 0.196i$
Analytic cond. $1.05402$
Root an. cond. $1.02665$
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  + (1.66 + 0.494i)3-s + (−0.393 + 0.127i)5-s + (1.02 − 1.41i)7-s + (2.51 + 1.64i)9-s + (−2.25 + 2.43i)11-s + (−2.14 − 0.697i)13-s + (−0.717 + 0.0178i)15-s + (−1.86 − 5.72i)17-s + (2.08 + 2.87i)19-s + (2.40 − 1.84i)21-s − 1.03i·23-s + (−3.90 + 2.83i)25-s + (3.35 + 3.96i)27-s + (−6.84 − 4.97i)29-s + (0.334 − 1.02i)31-s + ⋯
L(s)  = 1  + (0.958 + 0.285i)3-s + (−0.176 + 0.0572i)5-s + (0.388 − 0.534i)7-s + (0.837 + 0.546i)9-s + (−0.679 + 0.733i)11-s + (−0.595 − 0.193i)13-s + (−0.185 + 0.00460i)15-s + (−0.451 − 1.38i)17-s + (0.479 + 0.659i)19-s + (0.524 − 0.401i)21-s − 0.214i·23-s + (−0.781 + 0.567i)25-s + (0.646 + 0.762i)27-s + (−1.27 − 0.924i)29-s + (0.0599 − 0.184i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 132 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.196i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 132 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 - 0.196i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(132\)    =    \(2^{2} \cdot 3 \cdot 11\)
Sign: $0.980 - 0.196i$
Analytic conductor: \(1.05402\)
Root analytic conductor: \(1.02665\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{132} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 132,\ (\ :1/2),\ 0.980 - 0.196i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36366 + 0.135448i\)
\(L(\frac12)\) \(\approx\) \(1.36366 + 0.135448i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.66 - 0.494i)T \)
11 \( 1 + (2.25 - 2.43i)T \)
good5 \( 1 + (0.393 - 0.127i)T + (4.04 - 2.93i)T^{2} \)
7 \( 1 + (-1.02 + 1.41i)T + (-2.16 - 6.65i)T^{2} \)
13 \( 1 + (2.14 + 0.697i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (1.86 + 5.72i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-2.08 - 2.87i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 + 1.03iT - 23T^{2} \)
29 \( 1 + (6.84 + 4.97i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.334 + 1.02i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (6.18 + 4.49i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-2.22 + 1.61i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 2.87iT - 43T^{2} \)
47 \( 1 + (-3.87 - 5.32i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-6.60 - 2.14i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (6.47 - 8.91i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (-10.8 + 3.51i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 - 9.86T + 67T^{2} \)
71 \( 1 + (-15.3 + 4.99i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (-4.22 + 5.81i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (12.1 + 3.95i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (-3.25 - 10.0i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 7.11iT - 89T^{2} \)
97 \( 1 + (0.874 - 2.69i)T + (-78.4 - 57.0i)T^{2} \)
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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

−13.52617206309352760294921654300, −12.44259050662285757533200421716, −11.15023432952156334828169607053, −10.02389041251381427054474848282, −9.260481597317365033338852408950, −7.74719883467259516080051294791, −7.34562981870021851477493735320, −5.17464596586559494963684656201, −3.97257746700651579273299512815, −2.36248803373582592890742505134, 2.17736368998182740015645257139, 3.70586072471452449886932479939, 5.35934344867763706578459477245, 6.92961439011799150762582872008, 8.131133872387839182588973425962, 8.784944232843081815093976778701, 10.01764625692993344393231459629, 11.26534030715238398658888187380, 12.42369014692894082839902219838, 13.29104091212147254405973738763

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