Properties

Label 2-33-11.2-c6-0-3
Degree $2$
Conductor $33$
Sign $-0.105 - 0.994i$
Analytic cond. $7.59178$
Root an. cond. $2.75531$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (11.5 + 3.76i)2-s + (−12.6 + 9.16i)3-s + (68.3 + 49.6i)4-s + (42.9 + 132. i)5-s + (−180. + 58.7i)6-s + (−91.9 + 126. i)7-s + (147. + 202. i)8-s + (75.0 − 231. i)9-s + 1.69e3i·10-s + (569. + 1.20e3i)11-s − 1.31e3·12-s + (1.17e3 + 383. i)13-s + (−1.54e3 + 1.12e3i)14-s + (−1.75e3 − 1.27e3i)15-s + (−729. − 2.24e3i)16-s + (4.95e3 − 1.61e3i)17-s + ⋯
L(s)  = 1  + (1.44 + 0.470i)2-s + (−0.467 + 0.339i)3-s + (1.06 + 0.776i)4-s + (0.343 + 1.05i)5-s + (−0.836 + 0.271i)6-s + (−0.268 + 0.369i)7-s + (0.287 + 0.395i)8-s + (0.103 − 0.317i)9-s + 1.69i·10-s + (0.427 + 0.903i)11-s − 0.762·12-s + (0.537 + 0.174i)13-s + (−0.562 + 0.408i)14-s + (−0.518 − 0.377i)15-s + (−0.178 − 0.548i)16-s + (1.00 − 0.327i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.105 - 0.994i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.105 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $-0.105 - 0.994i$
Analytic conductor: \(7.59178\)
Root analytic conductor: \(2.75531\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :3),\ -0.105 - 0.994i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(2.05662 + 2.28692i\)
\(L(\frac12)\) \(\approx\) \(2.05662 + 2.28692i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (12.6 - 9.16i)T \)
11 \( 1 + (-569. - 1.20e3i)T \)
good2 \( 1 + (-11.5 - 3.76i)T + (51.7 + 37.6i)T^{2} \)
5 \( 1 + (-42.9 - 132. i)T + (-1.26e4 + 9.18e3i)T^{2} \)
7 \( 1 + (91.9 - 126. i)T + (-3.63e4 - 1.11e5i)T^{2} \)
13 \( 1 + (-1.17e3 - 383. i)T + (3.90e6 + 2.83e6i)T^{2} \)
17 \( 1 + (-4.95e3 + 1.61e3i)T + (1.95e7 - 1.41e7i)T^{2} \)
19 \( 1 + (5.98e3 + 8.23e3i)T + (-1.45e7 + 4.47e7i)T^{2} \)
23 \( 1 - 1.16e4T + 1.48e8T^{2} \)
29 \( 1 + (-2.39e3 + 3.30e3i)T + (-1.83e8 - 5.65e8i)T^{2} \)
31 \( 1 + (-5.58e3 + 1.71e4i)T + (-7.18e8 - 5.21e8i)T^{2} \)
37 \( 1 + (-2.99e4 - 2.17e4i)T + (7.92e8 + 2.44e9i)T^{2} \)
41 \( 1 + (5.22e4 + 7.19e4i)T + (-1.46e9 + 4.51e9i)T^{2} \)
43 \( 1 - 1.04e5iT - 6.32e9T^{2} \)
47 \( 1 + (1.13e5 - 8.26e4i)T + (3.33e9 - 1.02e10i)T^{2} \)
53 \( 1 + (-3.04e4 + 9.36e4i)T + (-1.79e10 - 1.30e10i)T^{2} \)
59 \( 1 + (-1.14e5 - 8.31e4i)T + (1.30e10 + 4.01e10i)T^{2} \)
61 \( 1 + (2.56e5 - 8.33e4i)T + (4.16e10 - 3.02e10i)T^{2} \)
67 \( 1 + 1.62e5T + 9.04e10T^{2} \)
71 \( 1 + (1.85e5 + 5.70e5i)T + (-1.03e11 + 7.52e10i)T^{2} \)
73 \( 1 + (-3.04e4 + 4.18e4i)T + (-4.67e10 - 1.43e11i)T^{2} \)
79 \( 1 + (-7.07e5 - 2.29e5i)T + (1.96e11 + 1.42e11i)T^{2} \)
83 \( 1 + (8.89e5 - 2.88e5i)T + (2.64e11 - 1.92e11i)T^{2} \)
89 \( 1 - 1.09e6T + 4.96e11T^{2} \)
97 \( 1 + (-9.97e4 + 3.06e5i)T + (-6.73e11 - 4.89e11i)T^{2} \)
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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

−15.24419996270990418133584299896, −14.74793497007901054981532597233, −13.48278478377325364910528773931, −12.29927433258291113720772885950, −11.08262806111136991079696508630, −9.578904997992064905610609127209, −6.98541441026748812292682234971, −6.11351519193532265458090322887, −4.61587166711772600857829495428, −2.96930383925237878632405149690, 1.28524944485038835023501199934, 3.63530996244115896763453214901, 5.21508395213582967966109743566, 6.25645337006746060591731392980, 8.540110173406830169554378627734, 10.52548984942264865338434516378, 11.85327904110098579291191632896, 12.80289366644385286600305035088, 13.50144110327216213662354144191, 14.67784627741773082857239683424

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