Properties

Label 2-33-33.32-c5-0-2
Degree $2$
Conductor $33$
Sign $-0.635 - 0.772i$
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  + 4.46·2-s + (−14.2 − 6.27i)3-s − 12.0·4-s + 59.8i·5-s + (−63.6 − 28.0i)6-s + 169. i·7-s − 196.·8-s + (164. + 179. i)9-s + 267. i·10-s + (−358. − 181. i)11-s + (172. + 75.8i)12-s − 838. i·13-s + 756. i·14-s + (375. − 854. i)15-s − 491.·16-s − 512.·17-s + ⋯
L(s)  = 1  + 0.788·2-s + (−0.915 − 0.402i)3-s − 0.377·4-s + 1.07i·5-s + (−0.722 − 0.317i)6-s + 1.30i·7-s − 1.08·8-s + (0.675 + 0.737i)9-s + 0.845i·10-s + (−0.892 − 0.451i)11-s + (0.345 + 0.152i)12-s − 1.37i·13-s + 1.03i·14-s + (0.431 − 0.980i)15-s − 0.479·16-s − 0.430·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.635 - 0.772i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.635 - 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $-0.635 - 0.772i$
Analytic conductor: \(5.29266\)
Root analytic conductor: \(2.30057\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :5/2),\ -0.635 - 0.772i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.339838 + 0.719495i\)
\(L(\frac12)\) \(\approx\) \(0.339838 + 0.719495i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (14.2 + 6.27i)T \)
11 \( 1 + (358. + 181. i)T \)
good2 \( 1 - 4.46T + 32T^{2} \)
5 \( 1 - 59.8iT - 3.12e3T^{2} \)
7 \( 1 - 169. iT - 1.68e4T^{2} \)
13 \( 1 + 838. iT - 3.71e5T^{2} \)
17 \( 1 + 512.T + 1.41e6T^{2} \)
19 \( 1 - 1.87e3iT - 2.47e6T^{2} \)
23 \( 1 - 3.33e3iT - 6.43e6T^{2} \)
29 \( 1 + 1.32e3T + 2.05e7T^{2} \)
31 \( 1 - 6.73e3T + 2.86e7T^{2} \)
37 \( 1 - 4.95e3T + 6.93e7T^{2} \)
41 \( 1 + 1.77e4T + 1.15e8T^{2} \)
43 \( 1 - 7.85e3iT - 1.47e8T^{2} \)
47 \( 1 - 2.01e4iT - 2.29e8T^{2} \)
53 \( 1 + 1.87e4iT - 4.18e8T^{2} \)
59 \( 1 - 5.84e3iT - 7.14e8T^{2} \)
61 \( 1 + 1.70e3iT - 8.44e8T^{2} \)
67 \( 1 + 2.93e4T + 1.35e9T^{2} \)
71 \( 1 - 1.73e4iT - 1.80e9T^{2} \)
73 \( 1 - 1.66e4iT - 2.07e9T^{2} \)
79 \( 1 + 2.58e3iT - 3.07e9T^{2} \)
83 \( 1 + 3.19e4T + 3.93e9T^{2} \)
89 \( 1 - 6.37e4iT - 5.58e9T^{2} \)
97 \( 1 - 1.03e5T + 8.58e9T^{2} \)
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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

−15.72570375536652832797913528601, −14.96633966045740790880808038286, −13.49741333753315922154670681763, −12.55041135970528924057654196027, −11.45083319514742308814079707816, −10.10973617248327058409789935913, −8.057879940906696116716634097272, −6.11108902749414972766624663584, −5.32486068657139785671111711772, −2.95622490059246783513194538887, 0.41839507253354455158098168910, 4.34579097973186112987807189854, 4.86313399842240998522389021378, 6.73882915619198983424819411975, 8.928112025174708980656032718048, 10.29060828328713455592751562036, 11.78144721812198888853070880831, 12.93131565500976581069933502478, 13.72634347685295915527451800431, 15.31725879644629946668272436833

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