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Properties

Label 2-33-33.32-c7-0-3
Degree $2$
Conductor $33$
Sign $-0.306 - 0.951i$
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  + 10.1·2-s + (−20.0 − 42.2i)3-s − 24.4·4-s + 419. i·5-s + (−203. − 429. i)6-s + 43.4i·7-s − 1.55e3·8-s + (−1.38e3 + 1.69e3i)9-s + 4.27e3i·10-s + (4.37e3 + 576. i)11-s + (490. + 1.03e3i)12-s + 1.30e4i·13-s + 442. i·14-s + (1.77e4 − 8.41e3i)15-s − 1.26e4·16-s − 3.33e4·17-s + ⋯
L(s)  = 1  + 0.899·2-s + (−0.428 − 0.903i)3-s − 0.191·4-s + 1.50i·5-s + (−0.385 − 0.812i)6-s + 0.0479i·7-s − 1.07·8-s + (−0.632 + 0.774i)9-s + 1.35i·10-s + (0.991 + 0.130i)11-s + (0.0818 + 0.172i)12-s + 1.65i·13-s + 0.0430i·14-s + (1.35 − 0.643i)15-s − 0.772·16-s − 1.64·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.306 - 0.951i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.306 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $-0.306 - 0.951i$
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: \((2,\ 33,\ (\ :7/2),\ -0.306 - 0.951i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.707324 + 0.971114i\)
\(L(\frac12)\) \(\approx\) \(0.707324 + 0.971114i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (20.0 + 42.2i)T \)
11 \( 1 + (-4.37e3 - 576. i)T \)
good2 \( 1 - 10.1T + 128T^{2} \)
5 \( 1 - 419. iT - 7.81e4T^{2} \)
7 \( 1 - 43.4iT - 8.23e5T^{2} \)
13 \( 1 - 1.30e4iT - 6.27e7T^{2} \)
17 \( 1 + 3.33e4T + 4.10e8T^{2} \)
19 \( 1 + 667. iT - 8.93e8T^{2} \)
23 \( 1 + 5.56e4iT - 3.40e9T^{2} \)
29 \( 1 + 9.31e4T + 1.72e10T^{2} \)
31 \( 1 + 1.11e5T + 2.75e10T^{2} \)
37 \( 1 - 4.87e5T + 9.49e10T^{2} \)
41 \( 1 + 3.51e5T + 1.94e11T^{2} \)
43 \( 1 - 5.35e5iT - 2.71e11T^{2} \)
47 \( 1 + 7.36e5iT - 5.06e11T^{2} \)
53 \( 1 - 1.13e6iT - 1.17e12T^{2} \)
59 \( 1 - 1.44e6iT - 2.48e12T^{2} \)
61 \( 1 - 3.61e5iT - 3.14e12T^{2} \)
67 \( 1 - 8.03e5T + 6.06e12T^{2} \)
71 \( 1 + 2.35e6iT - 9.09e12T^{2} \)
73 \( 1 - 2.26e6iT - 1.10e13T^{2} \)
79 \( 1 - 2.50e6iT - 1.92e13T^{2} \)
83 \( 1 - 2.38e6T + 2.71e13T^{2} \)
89 \( 1 - 4.69e6iT - 4.42e13T^{2} \)
97 \( 1 + 1.19e6T + 8.07e13T^{2} \)
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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

−14.93760020716487622582704205936, −14.20379968049968146523122700648, −13.31043868509843352701987698372, −11.87950329741483040592610439774, −11.09277712208000955180715438300, −9.062184221784305658597529106557, −6.90274375767376764496183360060, −6.28947134078004026520330655042, −4.22095137728656159718085910284, −2.34042108255282868658786099815, 0.43666706667710276992026237144, 3.76383139268196719619566428066, 4.86421748991726907051740415019, 5.84886437863275544583832721006, 8.652327869876658163408867863934, 9.506067776145530520426282405201, 11.34492006444500812034712919973, 12.55651707278133010471668709745, 13.39943197362265231059512187534, 14.94127539434254499772856198165

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