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Properties

Label 2-330-11.4-c1-0-2
Degree $2$
Conductor $330$
Sign $0.964 + 0.263i$
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 + (0.886 + 2.72i)7-s + (0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s + 10-s + (3.12 − 1.09i)11-s + 12-s + (4.25 + 3.09i)13-s + (0.886 − 2.72i)14-s + (0.309 + 0.951i)15-s + (−0.809 + 0.587i)16-s + (−0.125 + 0.0909i)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 + (0.335 + 1.03i)7-s + (0.109 − 0.336i)8-s + (−0.269 − 0.195i)9-s + 0.316·10-s + (0.943 − 0.331i)11-s + 0.288·12-s + (1.18 + 0.857i)13-s + (0.236 − 0.729i)14-s + (0.0797 + 0.245i)15-s + (−0.202 + 0.146i)16-s + (−0.0303 + 0.0220i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.263i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 + 0.263i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(330\)    =    \(2 \cdot 3 \cdot 5 \cdot 11\)
Sign: $0.964 + 0.263i$
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: \((2,\ 330,\ (\ :1/2),\ 0.964 + 0.263i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09843 - 0.147553i\)
\(L(\frac12)\) \(\approx\) \(1.09843 - 0.147553i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.12 + 1.09i)T \)
good7 \( 1 + (-0.886 - 2.72i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-4.25 - 3.09i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (0.125 - 0.0909i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.452 + 1.39i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 3.78T + 23T^{2} \)
29 \( 1 + (1.07 + 3.31i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (2.41 + 1.75i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.73 - 8.42i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-2.59 + 8.00i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 7.86T + 43T^{2} \)
47 \( 1 + (3.39 - 10.4i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (8.55 + 6.21i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-2.44 - 7.53i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (1.40 - 1.02i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 11.2T + 67T^{2} \)
71 \( 1 + (2.86 - 2.08i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (1.85 + 5.70i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-5.49 - 3.99i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-1.23 + 0.898i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 10.8T + 89T^{2} \)
97 \( 1 + (15.7 + 11.4i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.49491679429969103802655146590, −10.95238283892979815993324224003, −9.327273604894520127041765816543, −8.862320211400456243658931398552, −7.965236731546561535009605757500, −6.81084097258163407470437842994, −5.92686611817156328038300449033, −4.15517499694971029580306147374, −2.85604296542414846320732787313, −1.46148091737709959574661161453, 1.18293015792955567605190311920, 3.51203686311743178630289471281, 4.49604526325548448957536474643, 5.78562441477758387290308072625, 7.03429197311164416124107814091, 7.909303324863012373695243699848, 8.812532251353467455192381034649, 9.632014297824354451222270554451, 10.78689746364518544306711154942, 11.11787134959940746572769417936

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