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Properties

Label 2-330-11.7-c2-0-9
Degree $2$
Conductor $330$
Sign $-0.0216 + 0.999i$
Analytic cond. $8.99184$
Root an. cond. $2.99864$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.831 + 1.14i)2-s + (−0.535 + 1.64i)3-s + (−0.618 − 1.90i)4-s + (1.80 − 1.31i)5-s + (−1.43 − 1.98i)6-s + (−3.28 + 1.06i)7-s + (2.68 + 0.874i)8-s + (−2.42 − 1.76i)9-s + 3.16i·10-s + (−10.2 + 3.89i)11-s + 3.46·12-s + (6.91 − 9.51i)13-s + (1.50 − 4.64i)14-s + (1.19 + 3.68i)15-s + (−3.23 + 2.35i)16-s + (−5.01 − 6.90i)17-s + ⋯
L(s)  = 1  + (−0.415 + 0.572i)2-s + (−0.178 + 0.549i)3-s + (−0.154 − 0.475i)4-s + (0.361 − 0.262i)5-s + (−0.239 − 0.330i)6-s + (−0.468 + 0.152i)7-s + (0.336 + 0.109i)8-s + (−0.269 − 0.195i)9-s + 0.316i·10-s + (−0.935 + 0.354i)11-s + 0.288·12-s + (0.532 − 0.732i)13-s + (0.107 − 0.331i)14-s + (0.0797 + 0.245i)15-s + (−0.202 + 0.146i)16-s + (−0.295 − 0.406i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0216 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 330 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0216 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(330\)    =    \(2 \cdot 3 \cdot 5 \cdot 11\)
Sign: $-0.0216 + 0.999i$
Analytic conductor: \(8.99184\)
Root analytic conductor: \(2.99864\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{330} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 330,\ (\ :1),\ -0.0216 + 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.253437 - 0.258977i\)
\(L(\frac12)\) \(\approx\) \(0.253437 - 0.258977i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.831 - 1.14i)T \)
3 \( 1 + (0.535 - 1.64i)T \)
5 \( 1 + (-1.80 + 1.31i)T \)
11 \( 1 + (10.2 - 3.89i)T \)
good7 \( 1 + (3.28 - 1.06i)T + (39.6 - 28.8i)T^{2} \)
13 \( 1 + (-6.91 + 9.51i)T + (-52.2 - 160. i)T^{2} \)
17 \( 1 + (5.01 + 6.90i)T + (-89.3 + 274. i)T^{2} \)
19 \( 1 + (29.2 + 9.49i)T + (292. + 212. i)T^{2} \)
23 \( 1 + 17.3T + 529T^{2} \)
29 \( 1 + (-17.7 + 5.77i)T + (680. - 494. i)T^{2} \)
31 \( 1 + (0.810 + 0.588i)T + (296. + 913. i)T^{2} \)
37 \( 1 + (16.2 + 49.8i)T + (-1.10e3 + 804. i)T^{2} \)
41 \( 1 + (-6.82 - 2.21i)T + (1.35e3 + 988. i)T^{2} \)
43 \( 1 + 71.1iT - 1.84e3T^{2} \)
47 \( 1 + (-20.4 + 63.0i)T + (-1.78e3 - 1.29e3i)T^{2} \)
53 \( 1 + (-85.2 - 61.9i)T + (868. + 2.67e3i)T^{2} \)
59 \( 1 + (-18.6 - 57.5i)T + (-2.81e3 + 2.04e3i)T^{2} \)
61 \( 1 + (35.1 + 48.3i)T + (-1.14e3 + 3.53e3i)T^{2} \)
67 \( 1 + 129.T + 4.48e3T^{2} \)
71 \( 1 + (65.9 - 47.9i)T + (1.55e3 - 4.79e3i)T^{2} \)
73 \( 1 + (23.1 - 7.52i)T + (4.31e3 - 3.13e3i)T^{2} \)
79 \( 1 + (-36.2 + 49.8i)T + (-1.92e3 - 5.93e3i)T^{2} \)
83 \( 1 + (-8.69 - 11.9i)T + (-2.12e3 + 6.55e3i)T^{2} \)
89 \( 1 + 117.T + 7.92e3T^{2} \)
97 \( 1 + (21.1 + 15.3i)T + (2.90e3 + 8.94e3i)T^{2} \)
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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

−10.53175386283903847739128605668, −10.40049403350788947010112055108, −9.124431784022726316819589621795, −8.501952181623867470424238716958, −7.29462963218029301261188977111, −6.12723761728695222076425981659, −5.35349598239114616846406218499, −4.17129175028997146346325246604, −2.44064904888917925875716163979, −0.18927755169256163515479128062, 1.69715417812700204385521398393, 2.92770704585539913079344452641, 4.35420794302484071489923475776, 5.98800898908149283094449437667, 6.71500215597475152749433766951, 8.018216557819084651762289472629, 8.725395941484471231544119803187, 9.986626213741593272088221062025, 10.63531049383032646702754686404, 11.48090733873315192415514589381

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