Properties

Label 2-33-11.3-c5-0-8
Degree $2$
Conductor $33$
Sign $-0.633 + 0.774i$
Analytic cond. $5.29266$
Root an. cond. $2.30057$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (5.51 − 4.00i)2-s + (−2.78 − 8.55i)3-s + (4.46 − 13.7i)4-s + (−62.5 − 45.4i)5-s + (−49.6 − 36.0i)6-s + (59.3 − 182. i)7-s + (36.9 + 113. i)8-s + (−65.5 + 47.6i)9-s − 527.·10-s + (385. − 111. i)11-s − 129.·12-s + (−216. + 157. i)13-s + (−404. − 1.24e3i)14-s + (−215. + 662. i)15-s + (1.03e3 + 750. i)16-s + (1.18e3 + 864. i)17-s + ⋯
L(s)  = 1  + (0.974 − 0.708i)2-s + (−0.178 − 0.549i)3-s + (0.139 − 0.429i)4-s + (−1.11 − 0.813i)5-s + (−0.562 − 0.408i)6-s + (0.458 − 1.40i)7-s + (0.204 + 0.628i)8-s + (−0.269 + 0.195i)9-s − 1.66·10-s + (0.960 − 0.277i)11-s − 0.260·12-s + (−0.355 + 0.258i)13-s + (−0.551 − 1.69i)14-s + (−0.246 + 0.760i)15-s + (1.00 + 0.733i)16-s + (0.998 + 0.725i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.633 + 0.774i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.633 + 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $-0.633 + 0.774i$
Analytic conductor: \(5.29266\)
Root analytic conductor: \(2.30057\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :5/2),\ -0.633 + 0.774i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.860974 - 1.81633i\)
\(L(\frac12)\) \(\approx\) \(0.860974 - 1.81633i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.78 + 8.55i)T \)
11 \( 1 + (-385. + 111. i)T \)
good2 \( 1 + (-5.51 + 4.00i)T + (9.88 - 30.4i)T^{2} \)
5 \( 1 + (62.5 + 45.4i)T + (965. + 2.97e3i)T^{2} \)
7 \( 1 + (-59.3 + 182. i)T + (-1.35e4 - 9.87e3i)T^{2} \)
13 \( 1 + (216. - 157. i)T + (1.14e5 - 3.53e5i)T^{2} \)
17 \( 1 + (-1.18e3 - 864. i)T + (4.38e5 + 1.35e6i)T^{2} \)
19 \( 1 + (727. + 2.23e3i)T + (-2.00e6 + 1.45e6i)T^{2} \)
23 \( 1 - 2.22e3T + 6.43e6T^{2} \)
29 \( 1 + (2.11e3 - 6.51e3i)T + (-1.65e7 - 1.20e7i)T^{2} \)
31 \( 1 + (-4.80e3 + 3.48e3i)T + (8.84e6 - 2.72e7i)T^{2} \)
37 \( 1 + (-165. + 509. i)T + (-5.61e7 - 4.07e7i)T^{2} \)
41 \( 1 + (3.60e3 + 1.11e4i)T + (-9.37e7 + 6.80e7i)T^{2} \)
43 \( 1 + 1.26e4T + 1.47e8T^{2} \)
47 \( 1 + (-143. - 441. i)T + (-1.85e8 + 1.34e8i)T^{2} \)
53 \( 1 + (5.39e3 - 3.92e3i)T + (1.29e8 - 3.97e8i)T^{2} \)
59 \( 1 + (1.14e4 - 3.52e4i)T + (-5.78e8 - 4.20e8i)T^{2} \)
61 \( 1 + (-1.71e4 - 1.24e4i)T + (2.60e8 + 8.03e8i)T^{2} \)
67 \( 1 - 2.33e4T + 1.35e9T^{2} \)
71 \( 1 + (-3.54e4 - 2.57e4i)T + (5.57e8 + 1.71e9i)T^{2} \)
73 \( 1 + (8.74e3 - 2.69e4i)T + (-1.67e9 - 1.21e9i)T^{2} \)
79 \( 1 + (-3.99e4 + 2.90e4i)T + (9.50e8 - 2.92e9i)T^{2} \)
83 \( 1 + (-3.41e4 - 2.48e4i)T + (1.21e9 + 3.74e9i)T^{2} \)
89 \( 1 - 8.88e3T + 5.58e9T^{2} \)
97 \( 1 + (9.30e4 - 6.76e4i)T + (2.65e9 - 8.16e9i)T^{2} \)
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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

−14.83108641644962277279338895567, −13.73165220282344697340968446711, −12.71630503362554271850054838717, −11.76389477555670431541094706710, −10.89268024330373025709130199172, −8.485200543553834057807744635922, −7.20190538926814750690249378957, −4.80842835004593528787932213581, −3.74804262059930843436450034125, −1.05573402878071172047895799736, 3.47466568352350916334579490898, 4.98419618189519350082388573618, 6.37739140955897831615941041800, 7.949786978419292794730807811603, 9.807175227872369746410129834216, 11.57020747177017006154329530871, 12.29853987402347090434984204375, 14.36636472322729903375887403601, 14.96371981275016348163957906042, 15.56751360742167170315756221331

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