Properties

Label 2-33-33.32-c7-0-17
Degree $2$
Conductor $33$
Sign $-0.468 + 0.883i$
Analytic cond. $10.3087$
Root an. cond. $3.21071$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 15.1·2-s + (46.1 − 7.41i)3-s + 101.·4-s − 382. i·5-s + (−698. + 112. i)6-s + 606. i·7-s + 408.·8-s + (2.07e3 − 684. i)9-s + 5.78e3i·10-s + (−2.66e3 + 3.52e3i)11-s + (4.66e3 − 749. i)12-s − 1.19e4i·13-s − 9.17e3i·14-s + (−2.83e3 − 1.76e4i)15-s − 1.91e4·16-s − 1.15e4·17-s + ⋯
L(s)  = 1  − 1.33·2-s + (0.987 − 0.158i)3-s + 0.789·4-s − 1.36i·5-s + (−1.32 + 0.212i)6-s + 0.667i·7-s + 0.281·8-s + (0.949 − 0.313i)9-s + 1.82i·10-s + (−0.602 + 0.797i)11-s + (0.779 − 0.125i)12-s − 1.51i·13-s − 0.893i·14-s + (−0.216 − 1.35i)15-s − 1.16·16-s − 0.570·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.468 + 0.883i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.468 + 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $-0.468 + 0.883i$
Analytic conductor: \(10.3087\)
Root analytic conductor: \(3.21071\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :7/2),\ -0.468 + 0.883i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.466786 - 0.776105i\)
\(L(\frac12)\) \(\approx\) \(0.466786 - 0.776105i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-46.1 + 7.41i)T \)
11 \( 1 + (2.66e3 - 3.52e3i)T \)
good2 \( 1 + 15.1T + 128T^{2} \)
5 \( 1 + 382. iT - 7.81e4T^{2} \)
7 \( 1 - 606. iT - 8.23e5T^{2} \)
13 \( 1 + 1.19e4iT - 6.27e7T^{2} \)
17 \( 1 + 1.15e4T + 4.10e8T^{2} \)
19 \( 1 + 3.96e4iT - 8.93e8T^{2} \)
23 \( 1 + 6.51e4iT - 3.40e9T^{2} \)
29 \( 1 + 1.06e5T + 1.72e10T^{2} \)
31 \( 1 + 5.03e4T + 2.75e10T^{2} \)
37 \( 1 + 4.87e5T + 9.49e10T^{2} \)
41 \( 1 - 3.86e5T + 1.94e11T^{2} \)
43 \( 1 + 3.85e5iT - 2.71e11T^{2} \)
47 \( 1 - 5.78e5iT - 5.06e11T^{2} \)
53 \( 1 + 4.24e5iT - 1.17e12T^{2} \)
59 \( 1 + 9.79e5iT - 2.48e12T^{2} \)
61 \( 1 - 3.28e6iT - 3.14e12T^{2} \)
67 \( 1 - 9.90e5T + 6.06e12T^{2} \)
71 \( 1 - 1.26e5iT - 9.09e12T^{2} \)
73 \( 1 + 1.59e6iT - 1.10e13T^{2} \)
79 \( 1 + 3.65e6iT - 1.92e13T^{2} \)
83 \( 1 - 9.71e6T + 2.71e13T^{2} \)
89 \( 1 + 1.86e6iT - 4.42e13T^{2} \)
97 \( 1 - 4.20e6T + 8.07e13T^{2} \)
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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

−15.23118456086735497946677623879, −13.26940486014670310643318409281, −12.54485658662070299521598709135, −10.43192094529551627846233032994, −9.140950538468139500337809114065, −8.580732911535450394817054180884, −7.46140214090268174274058301543, −4.85640744321364054001488779784, −2.23674648544945236017776357527, −0.57701077882646456002324230874, 1.91015306740598997809035551020, 3.70975706589759108487218430212, 6.90932347653775479167961600022, 7.83598803285733387023521999367, 9.187810372514279151671574508718, 10.29172582365827552882298417911, 11.13796574905566350515965647791, 13.62818538696295047909373783796, 14.30246117706989341740410842291, 15.75028908215714493105644919190

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