Properties

Label 2-33-11.4-c5-0-4
Degree $2$
Conductor $33$
Sign $-0.288 - 0.957i$
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  + (8.77 + 6.37i)2-s + (−2.78 + 8.55i)3-s + (26.4 + 81.4i)4-s + (31.3 − 22.8i)5-s + (−78.9 + 57.3i)6-s + (−67.6 − 208. i)7-s + (−179. + 553. i)8-s + (−65.5 − 47.6i)9-s + 420.·10-s + (313. + 250. i)11-s − 770.·12-s + (32.8 + 23.8i)13-s + (734. − 2.25e3i)14-s + (107. + 332. i)15-s + (−2.89e3 + 2.09e3i)16-s + (516. − 375. i)17-s + ⋯
L(s)  = 1  + (1.55 + 1.12i)2-s + (−0.178 + 0.549i)3-s + (0.827 + 2.54i)4-s + (0.561 − 0.407i)5-s + (−0.895 + 0.650i)6-s + (−0.522 − 1.60i)7-s + (−0.993 + 3.05i)8-s + (−0.269 − 0.195i)9-s + 1.33·10-s + (0.781 + 0.624i)11-s − 1.54·12-s + (0.0539 + 0.0391i)13-s + (1.00 − 3.08i)14-s + (0.123 + 0.381i)15-s + (−2.82 + 2.05i)16-s + (0.433 − 0.315i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.288 - 0.957i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.288 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $-0.288 - 0.957i$
Analytic conductor: \(5.29266\)
Root analytic conductor: \(2.30057\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :5/2),\ -0.288 - 0.957i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.97015 + 2.65011i\)
\(L(\frac12)\) \(\approx\) \(1.97015 + 2.65011i\)
\(L(\frac{7}{2})\) 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 \( 1 + (-313. - 250. i)T \)
good2 \( 1 + (-8.77 - 6.37i)T + (9.88 + 30.4i)T^{2} \)
5 \( 1 + (-31.3 + 22.8i)T + (965. - 2.97e3i)T^{2} \)
7 \( 1 + (67.6 + 208. i)T + (-1.35e4 + 9.87e3i)T^{2} \)
13 \( 1 + (-32.8 - 23.8i)T + (1.14e5 + 3.53e5i)T^{2} \)
17 \( 1 + (-516. + 375. i)T + (4.38e5 - 1.35e6i)T^{2} \)
19 \( 1 + (-427. + 1.31e3i)T + (-2.00e6 - 1.45e6i)T^{2} \)
23 \( 1 + 799.T + 6.43e6T^{2} \)
29 \( 1 + (1.31e3 + 4.06e3i)T + (-1.65e7 + 1.20e7i)T^{2} \)
31 \( 1 + (4.93e3 + 3.58e3i)T + (8.84e6 + 2.72e7i)T^{2} \)
37 \( 1 + (-3.01e3 - 9.28e3i)T + (-5.61e7 + 4.07e7i)T^{2} \)
41 \( 1 + (1.35e3 - 4.15e3i)T + (-9.37e7 - 6.80e7i)T^{2} \)
43 \( 1 - 2.01e3T + 1.47e8T^{2} \)
47 \( 1 + (-110. + 339. i)T + (-1.85e8 - 1.34e8i)T^{2} \)
53 \( 1 + (2.87e4 + 2.09e4i)T + (1.29e8 + 3.97e8i)T^{2} \)
59 \( 1 + (3.66e3 + 1.12e4i)T + (-5.78e8 + 4.20e8i)T^{2} \)
61 \( 1 + (2.27e4 - 1.65e4i)T + (2.60e8 - 8.03e8i)T^{2} \)
67 \( 1 - 2.13e4T + 1.35e9T^{2} \)
71 \( 1 + (1.96e3 - 1.43e3i)T + (5.57e8 - 1.71e9i)T^{2} \)
73 \( 1 + (-2.07e4 - 6.39e4i)T + (-1.67e9 + 1.21e9i)T^{2} \)
79 \( 1 + (2.67e4 + 1.94e4i)T + (9.50e8 + 2.92e9i)T^{2} \)
83 \( 1 + (-5.48e4 + 3.98e4i)T + (1.21e9 - 3.74e9i)T^{2} \)
89 \( 1 - 6.97e4T + 5.58e9T^{2} \)
97 \( 1 + (-1.36e5 - 9.93e4i)T + (2.65e9 + 8.16e9i)T^{2} \)
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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

−15.97251899926538482693531731741, −14.73184411195963025391795743714, −13.72551105515882881594652208615, −12.97831839765311029313713102095, −11.48385868528780940638586886149, −9.606789603679782829316668372728, −7.50151965856367027027985678958, −6.34686521071597273594170172691, −4.80732499883956788025574233999, −3.69048413073264882110193497994, 1.86745882159455607932831313307, 3.27037667779722607848580689472, 5.61517210662983089770365098198, 6.26110860517633655680785760830, 9.326592321951313205687213551515, 10.80911287123123901892669122641, 12.08310187471102151401652161844, 12.60218709577177589110968831037, 13.97968747583757649238949832447, 14.70227236879531136175518029454

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