Properties

Label 2-33-11.3-c5-0-1
Degree $2$
Conductor $33$
Sign $-0.455 - 0.890i$
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  + (0.319 − 0.231i)2-s + (−2.78 − 8.55i)3-s + (−9.84 + 30.2i)4-s + (−17.3 − 12.5i)5-s + (−2.87 − 2.08i)6-s + (−52.3 + 161. i)7-s + (7.77 + 23.9i)8-s + (−65.5 + 47.6i)9-s − 8.43·10-s + (−52.4 + 397. i)11-s + 286.·12-s + (−256. + 186. i)13-s + (20.6 + 63.4i)14-s + (−59.4 + 183. i)15-s + (−816. − 593. i)16-s + (−406. − 295. i)17-s + ⋯
L(s)  = 1  + (0.0563 − 0.0409i)2-s + (−0.178 − 0.549i)3-s + (−0.307 + 0.946i)4-s + (−0.309 − 0.224i)5-s + (−0.0325 − 0.0236i)6-s + (−0.403 + 1.24i)7-s + (0.0429 + 0.132i)8-s + (−0.269 + 0.195i)9-s − 0.0266·10-s + (−0.130 + 0.991i)11-s + 0.574·12-s + (−0.421 + 0.306i)13-s + (0.0281 + 0.0865i)14-s + (−0.0682 + 0.210i)15-s + (−0.797 − 0.579i)16-s + (−0.341 − 0.247i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.455 - 0.890i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.455 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $-0.455 - 0.890i$
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: \((2,\ 33,\ (\ :5/2),\ -0.455 - 0.890i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.419732 + 0.686578i\)
\(L(\frac12)\) \(\approx\) \(0.419732 + 0.686578i\)
\(L(\frac{7}{2})\) 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 \( 1 + (52.4 - 397. i)T \)
good2 \( 1 + (-0.319 + 0.231i)T + (9.88 - 30.4i)T^{2} \)
5 \( 1 + (17.3 + 12.5i)T + (965. + 2.97e3i)T^{2} \)
7 \( 1 + (52.3 - 161. i)T + (-1.35e4 - 9.87e3i)T^{2} \)
13 \( 1 + (256. - 186. i)T + (1.14e5 - 3.53e5i)T^{2} \)
17 \( 1 + (406. + 295. i)T + (4.38e5 + 1.35e6i)T^{2} \)
19 \( 1 + (53.6 + 165. i)T + (-2.00e6 + 1.45e6i)T^{2} \)
23 \( 1 - 3.67e3T + 6.43e6T^{2} \)
29 \( 1 + (-810. + 2.49e3i)T + (-1.65e7 - 1.20e7i)T^{2} \)
31 \( 1 + (2.05e3 - 1.49e3i)T + (8.84e6 - 2.72e7i)T^{2} \)
37 \( 1 + (3.87e3 - 1.19e4i)T + (-5.61e7 - 4.07e7i)T^{2} \)
41 \( 1 + (-5.96e3 - 1.83e4i)T + (-9.37e7 + 6.80e7i)T^{2} \)
43 \( 1 + 6.55e3T + 1.47e8T^{2} \)
47 \( 1 + (-2.39e3 - 7.38e3i)T + (-1.85e8 + 1.34e8i)T^{2} \)
53 \( 1 + (-1.39e3 + 1.01e3i)T + (1.29e8 - 3.97e8i)T^{2} \)
59 \( 1 + (-7.12e3 + 2.19e4i)T + (-5.78e8 - 4.20e8i)T^{2} \)
61 \( 1 + (-3.62e4 - 2.63e4i)T + (2.60e8 + 8.03e8i)T^{2} \)
67 \( 1 - 2.33e4T + 1.35e9T^{2} \)
71 \( 1 + (5.24e4 + 3.80e4i)T + (5.57e8 + 1.71e9i)T^{2} \)
73 \( 1 + (1.79e4 - 5.53e4i)T + (-1.67e9 - 1.21e9i)T^{2} \)
79 \( 1 + (-6.60e4 + 4.80e4i)T + (9.50e8 - 2.92e9i)T^{2} \)
83 \( 1 + (-8.07e4 - 5.86e4i)T + (1.21e9 + 3.74e9i)T^{2} \)
89 \( 1 + 6.83e4T + 5.58e9T^{2} \)
97 \( 1 + (1.00e5 - 7.29e4i)T + (2.65e9 - 8.16e9i)T^{2} \)
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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

−16.16087862941070284434781534795, −14.96692583552952378617674203592, −13.25421599970085020002088762232, −12.40058755970377785089252064546, −11.65601531591929943878181550593, −9.421618526254691478299220436831, −8.232013219390280972877186750320, −6.82540220344629839683260263284, −4.81436695335242413930651193701, −2.63014646974166048240234032958, 0.48619982091093134518050593542, 3.76709273861909836810294160293, 5.41440888518040575216234593648, 7.05226583010772619126778610163, 9.055612441272069279667404964227, 10.42776325436146369921552205002, 11.04768573459058309015270153372, 13.13186668070183007479108859795, 14.19733263505259348212608946989, 15.26683748492378789185165558606

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