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Properties

Label 2-33-3.2-c4-0-11
Degree $2$
Conductor $33$
Sign $-0.866 + 0.499i$
Analytic cond. $3.41120$
Root an. cond. $1.84694$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.08i·2-s + (−4.49 − 7.79i)3-s + 6.45·4-s − 9.71i·5-s + (−24.0 + 13.8i)6-s − 66.8·7-s − 69.3i·8-s + (−40.5 + 70.1i)9-s − 30.0·10-s − 36.4i·11-s + (−29.0 − 50.3i)12-s + 305.·13-s + 206. i·14-s + (−75.7 + 43.7i)15-s − 110.·16-s − 433. i·17-s + ⋯
L(s)  = 1  − 0.772i·2-s + (−0.499 − 0.866i)3-s + 0.403·4-s − 0.388i·5-s + (−0.668 + 0.385i)6-s − 1.36·7-s − 1.08i·8-s + (−0.500 + 0.865i)9-s − 0.300·10-s − 0.301i·11-s + (−0.201 − 0.349i)12-s + 1.80·13-s + 1.05i·14-s + (−0.336 + 0.194i)15-s − 0.433·16-s − 1.49i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 + 0.499i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.866 + 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $-0.866 + 0.499i$
Analytic conductor: \(3.41120\)
Root analytic conductor: \(1.84694\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :2),\ -0.866 + 0.499i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.304733 - 1.13767i\)
\(L(\frac12)\) \(\approx\) \(0.304733 - 1.13767i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (4.49 + 7.79i)T \)
11 \( 1 + 36.4iT \)
good2 \( 1 + 3.08iT - 16T^{2} \)
5 \( 1 + 9.71iT - 625T^{2} \)
7 \( 1 + 66.8T + 2.40e3T^{2} \)
13 \( 1 - 305.T + 2.85e4T^{2} \)
17 \( 1 + 433. iT - 8.35e4T^{2} \)
19 \( 1 - 199.T + 1.30e5T^{2} \)
23 \( 1 - 267. iT - 2.79e5T^{2} \)
29 \( 1 - 278. iT - 7.07e5T^{2} \)
31 \( 1 + 885.T + 9.23e5T^{2} \)
37 \( 1 - 83.7T + 1.87e6T^{2} \)
41 \( 1 + 6.14iT - 2.82e6T^{2} \)
43 \( 1 - 854.T + 3.41e6T^{2} \)
47 \( 1 - 2.06e3iT - 4.87e6T^{2} \)
53 \( 1 - 379. iT - 7.89e6T^{2} \)
59 \( 1 - 6.22e3iT - 1.21e7T^{2} \)
61 \( 1 - 3.45e3T + 1.38e7T^{2} \)
67 \( 1 + 1.91e3T + 2.01e7T^{2} \)
71 \( 1 + 5.65e3iT - 2.54e7T^{2} \)
73 \( 1 + 2.38e3T + 2.83e7T^{2} \)
79 \( 1 + 3.67e3T + 3.89e7T^{2} \)
83 \( 1 + 8.45e3iT - 4.74e7T^{2} \)
89 \( 1 - 1.47e4iT - 6.27e7T^{2} \)
97 \( 1 - 1.57e4T + 8.85e7T^{2} \)
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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

−16.09869463293577193289896301796, −13.58738268179485035642264213881, −12.88076594291222028287352160302, −11.76851957914916297393964407161, −10.76749647486497903614114981737, −9.166523694339949772447594861634, −7.14277195497779118375043174016, −5.96909011801387393454312951260, −3.14130691068360885392875755281, −0.954783677693226214468085483630, 3.53631342676135792942963530382, 5.87077615201422677307866395122, 6.65755359175817586955600424611, 8.684889066126966658938103619450, 10.26876241849331995600623163041, 11.22810397552885208395013485278, 12.83435410249196118763974910481, 14.54105720162186208913620789405, 15.63019037370411676997895067570, 16.18312955070879582075046179054

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