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Properties

Label 2-330-11.4-c1-0-5
Degree $2$
Conductor $330$
Sign $0.530 + 0.847i$
Analytic cond. $2.63506$
Root an. cond. $1.62328$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.809 − 0.587i)2-s + (−0.309 + 0.951i)3-s + (0.309 + 0.951i)4-s + (0.809 − 0.587i)5-s + (0.809 − 0.587i)6-s + (−1 − 3.07i)7-s + (0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s − 10-s + (3.04 + 1.31i)11-s − 12-s + (−2.73 − 1.98i)13-s + (−1 + 3.07i)14-s + (0.309 + 0.951i)15-s + (−0.809 + 0.587i)16-s + (4.92 − 3.57i)17-s + ⋯
L(s)  = 1  + (−0.572 − 0.415i)2-s + (−0.178 + 0.549i)3-s + (0.154 + 0.475i)4-s + (0.361 − 0.262i)5-s + (0.330 − 0.239i)6-s + (−0.377 − 1.16i)7-s + (0.109 − 0.336i)8-s + (−0.269 − 0.195i)9-s − 0.316·10-s + (0.918 + 0.396i)11-s − 0.288·12-s + (−0.758 − 0.551i)13-s + (−0.267 + 0.822i)14-s + (0.0797 + 0.245i)15-s + (−0.202 + 0.146i)16-s + (1.19 − 0.868i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(330\)    =    \(2 \cdot 3 \cdot 5 \cdot 11\)
Sign: $0.530 + 0.847i$
Analytic conductor: \(2.63506\)
Root analytic conductor: \(1.62328\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{330} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 330,\ (\ :1/2),\ 0.530 + 0.847i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.841158 - 0.466108i\)
\(L(\frac12)\) \(\approx\) \(0.841158 - 0.466108i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 + 0.587i)T \)
3 \( 1 + (0.309 - 0.951i)T \)
5 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 + (-3.04 - 1.31i)T \)
good7 \( 1 + (1 + 3.07i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (2.73 + 1.98i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-4.92 + 3.57i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.381 + 1.17i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 6.85T + 23T^{2} \)
29 \( 1 + (0.809 + 2.48i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-1.11 - 0.812i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (1.97 + 6.06i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-0.381 + 1.17i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 0.0901T + 43T^{2} \)
47 \( 1 + (1.33 - 4.11i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (6.85 + 4.97i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-4.35 - 13.4i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 5.09T + 67T^{2} \)
71 \( 1 + (8.47 - 6.15i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-4.14 - 12.7i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-3.30 - 2.40i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (3.23 - 2.35i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 17.2T + 89T^{2} \)
97 \( 1 + (15.0 + 10.9i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.26603363051978341221040046737, −10.28473881986546970609345228030, −9.731965985477299149183196941290, −9.007096783082925707193462122342, −7.58006281429738948417772431548, −6.85790833163126549819217649238, −5.33554577798137018527696910163, −4.17706286804486207038170550852, −2.98300796374035956158616815583, −0.936447767184423893887300194954, 1.61807835429464512134003720748, 3.11140913679778807329830191461, 5.16966600708629897219699352910, 6.13282115157734824177470152705, 6.79453812073690855823578665174, 7.965956772469849319884819388251, 8.983650284425744557287640523569, 9.611459607682684373484301817151, 10.74876599119282854228473690453, 11.84966378252966298335183516819

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