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Properties

Label 2-33-3.2-c4-0-12
Degree $2$
Conductor $33$
Sign $-0.986 - 0.164i$
Analytic cond. $3.41120$
Root an. cond. $1.84694$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7.08i·2-s + (1.47 − 8.87i)3-s − 34.1·4-s + 5.51i·5-s + (−62.8 − 10.4i)6-s + 86.8·7-s + 128. i·8-s + (−76.6 − 26.2i)9-s + 39.1·10-s − 36.4i·11-s + (−50.5 + 303. i)12-s − 166.·13-s − 615. i·14-s + (48.9 + 8.16i)15-s + 365.·16-s − 57.8i·17-s + ⋯
L(s)  = 1  − 1.77i·2-s + (0.164 − 0.986i)3-s − 2.13·4-s + 0.220i·5-s + (−1.74 − 0.291i)6-s + 1.77·7-s + 2.01i·8-s + (−0.945 − 0.324i)9-s + 0.391·10-s − 0.301i·11-s + (−0.351 + 2.10i)12-s − 0.987·13-s − 3.13i·14-s + (0.217 + 0.0362i)15-s + 1.42·16-s − 0.200i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 - 0.164i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.986 - 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $-0.986 - 0.164i$
Analytic conductor: \(3.41120\)
Root analytic conductor: \(1.84694\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :2),\ -0.986 - 0.164i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.115680 + 1.39803i\)
\(L(\frac12)\) \(\approx\) \(0.115680 + 1.39803i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.47 + 8.87i)T \)
11 \( 1 + 36.4iT \)
good2 \( 1 + 7.08iT - 16T^{2} \)
5 \( 1 - 5.51iT - 625T^{2} \)
7 \( 1 - 86.8T + 2.40e3T^{2} \)
13 \( 1 + 166.T + 2.85e4T^{2} \)
17 \( 1 + 57.8iT - 8.35e4T^{2} \)
19 \( 1 - 469.T + 1.30e5T^{2} \)
23 \( 1 + 403. iT - 2.79e5T^{2} \)
29 \( 1 - 583. iT - 7.07e5T^{2} \)
31 \( 1 - 334.T + 9.23e5T^{2} \)
37 \( 1 - 793.T + 1.87e6T^{2} \)
41 \( 1 + 1.52e3iT - 2.82e6T^{2} \)
43 \( 1 + 578.T + 3.41e6T^{2} \)
47 \( 1 - 3.24e3iT - 4.87e6T^{2} \)
53 \( 1 - 4.29e3iT - 7.89e6T^{2} \)
59 \( 1 + 2.28e3iT - 1.21e7T^{2} \)
61 \( 1 + 4.99e3T + 1.38e7T^{2} \)
67 \( 1 - 1.34e3T + 2.01e7T^{2} \)
71 \( 1 + 421. iT - 2.54e7T^{2} \)
73 \( 1 + 1.40e3T + 2.83e7T^{2} \)
79 \( 1 + 3.50e3T + 3.89e7T^{2} \)
83 \( 1 + 1.46e3iT - 4.74e7T^{2} \)
89 \( 1 + 694. iT - 6.27e7T^{2} \)
97 \( 1 - 6.70e3T + 8.85e7T^{2} \)
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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

−14.51739161736543155101191395508, −13.91483957774634267659714337843, −12.46590683292595657125632267369, −11.65641961919487507495962132874, −10.74238333882979069680244276460, −8.991175921983977686290281717822, −7.66805902908306035739938305770, −4.92995144849624875494967281955, −2.67953471915558146886243651448, −1.19492086520459920349295247127, 4.60762930465750402776610778399, 5.35268807203762996271063650821, 7.52383277091980493495969266931, 8.477719845484936829755960474679, 9.767018187002247978171979275803, 11.57654283617952192881778103965, 13.77113392360506935491475296460, 14.70500310112039505215276060941, 15.23042849477006482795058590292, 16.50707972185215565129944839750

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