Properties

Label 2-33-11.9-c5-0-0
Degree $2$
Conductor $33$
Sign $-0.957 + 0.290i$
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  + (−1.74 + 5.36i)2-s + (−7.28 + 5.29i)3-s + (0.152 + 0.110i)4-s + (21.0 + 64.7i)5-s + (−15.6 − 48.2i)6-s + (−106. − 77.4i)7-s + (−146. + 106. i)8-s + (25.0 − 77.0i)9-s − 384.·10-s + (−98.6 − 388. i)11-s − 1.69·12-s + (−4.06 + 12.5i)13-s + (600. − 436. i)14-s + (−496. − 360. i)15-s + (−314. − 968. i)16-s + (618. + 1.90e3i)17-s + ⋯
L(s)  = 1  + (−0.308 + 0.948i)2-s + (−0.467 + 0.339i)3-s + (0.00476 + 0.00346i)4-s + (0.376 + 1.15i)5-s + (−0.177 − 0.547i)6-s + (−0.821 − 0.597i)7-s + (−0.811 + 0.589i)8-s + (0.103 − 0.317i)9-s − 1.21·10-s + (−0.245 − 0.969i)11-s − 0.00339·12-s + (−0.00667 + 0.0205i)13-s + (0.819 − 0.595i)14-s + (−0.569 − 0.413i)15-s + (−0.307 − 0.945i)16-s + (0.519 + 1.59i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 + 0.290i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.957 + 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $-0.957 + 0.290i$
Analytic conductor: \(5.29266\)
Root analytic conductor: \(2.30057\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :5/2),\ -0.957 + 0.290i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.123662 - 0.834458i\)
\(L(\frac12)\) \(\approx\) \(0.123662 - 0.834458i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (7.28 - 5.29i)T \)
11 \( 1 + (98.6 + 388. i)T \)
good2 \( 1 + (1.74 - 5.36i)T + (-25.8 - 18.8i)T^{2} \)
5 \( 1 + (-21.0 - 64.7i)T + (-2.52e3 + 1.83e3i)T^{2} \)
7 \( 1 + (106. + 77.4i)T + (5.19e3 + 1.59e4i)T^{2} \)
13 \( 1 + (4.06 - 12.5i)T + (-3.00e5 - 2.18e5i)T^{2} \)
17 \( 1 + (-618. - 1.90e3i)T + (-1.14e6 + 8.34e5i)T^{2} \)
19 \( 1 + (2.25e3 - 1.63e3i)T + (7.65e5 - 2.35e6i)T^{2} \)
23 \( 1 - 2.93e3T + 6.43e6T^{2} \)
29 \( 1 + (-1.92e3 - 1.40e3i)T + (6.33e6 + 1.95e7i)T^{2} \)
31 \( 1 + (603. - 1.85e3i)T + (-2.31e7 - 1.68e7i)T^{2} \)
37 \( 1 + (-7.39e3 - 5.37e3i)T + (2.14e7 + 6.59e7i)T^{2} \)
41 \( 1 + (7.64e3 - 5.55e3i)T + (3.58e7 - 1.10e8i)T^{2} \)
43 \( 1 + 1.94e3T + 1.47e8T^{2} \)
47 \( 1 + (9.32e3 - 6.77e3i)T + (7.08e7 - 2.18e8i)T^{2} \)
53 \( 1 + (5.78e3 - 1.77e4i)T + (-3.38e8 - 2.45e8i)T^{2} \)
59 \( 1 + (-3.67e4 - 2.67e4i)T + (2.20e8 + 6.79e8i)T^{2} \)
61 \( 1 + (1.06e4 + 3.26e4i)T + (-6.83e8 + 4.96e8i)T^{2} \)
67 \( 1 - 3.79e4T + 1.35e9T^{2} \)
71 \( 1 + (7.49e3 + 2.30e4i)T + (-1.45e9 + 1.06e9i)T^{2} \)
73 \( 1 + (-4.77e4 - 3.47e4i)T + (6.40e8 + 1.97e9i)T^{2} \)
79 \( 1 + (-1.30e4 + 4.00e4i)T + (-2.48e9 - 1.80e9i)T^{2} \)
83 \( 1 + (-5.75e3 - 1.77e4i)T + (-3.18e9 + 2.31e9i)T^{2} \)
89 \( 1 + 8.21e4T + 5.58e9T^{2} \)
97 \( 1 + (-3.62e4 + 1.11e5i)T + (-6.94e9 - 5.04e9i)T^{2} \)
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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

−16.61243710162430495993512487724, −15.24602334497214476929415451021, −14.41373836204648274529925302504, −12.79421818857777573840824813942, −11.01402761161117412472496077872, −10.16575229954841573722497579385, −8.319622884941196527539879323349, −6.68802934701342808366375841917, −6.04387955383525295037384864697, −3.29992255338462840500113835969, 0.57332494550074734156105506954, 2.39152381824872033114596091198, 5.09837921219479212205065532068, 6.73788899602552243658016555929, 9.017481002719024985166766864063, 9.870056178306505187238023416324, 11.41518000630249905578047495748, 12.55714894480814100872919919350, 13.05423431652118037200037720995, 15.26508898970995673787067283209

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