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Properties

Label 2-3330-185.184-c1-0-15
Degree $2$
Conductor $3330$
Sign $-0.931 - 0.363i$
Analytic cond. $26.5901$
Root an. cond. $5.15656$
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  + 2-s + 4-s + (0.0603 + 2.23i)5-s + 3.18i·7-s + 8-s + (0.0603 + 2.23i)10-s − 2.65·11-s − 0.269·13-s + 3.18i·14-s + 16-s − 1.00·17-s + 0.921i·19-s + (0.0603 + 2.23i)20-s − 2.65·22-s − 2.37·23-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + (0.0269 + 0.999i)5-s + 1.20i·7-s + 0.353·8-s + (0.0190 + 0.706i)10-s − 0.800·11-s − 0.0747·13-s + 0.850i·14-s + 0.250·16-s − 0.244·17-s + 0.211i·19-s + (0.0134 + 0.499i)20-s − 0.565·22-s − 0.495·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.931 - 0.363i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.931 - 0.363i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3330\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 37\)
Sign: $-0.931 - 0.363i$
Analytic conductor: \(26.5901\)
Root analytic conductor: \(5.15656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3330} (739, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3330,\ (\ :1/2),\ -0.931 - 0.363i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.752585634\)
\(L(\frac12)\) \(\approx\) \(1.752585634\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
5 \( 1 + (-0.0603 - 2.23i)T \)
37 \( 1 + (2.36 - 5.60i)T \)
good7 \( 1 - 3.18iT - 7T^{2} \)
11 \( 1 + 2.65T + 11T^{2} \)
13 \( 1 + 0.269T + 13T^{2} \)
17 \( 1 + 1.00T + 17T^{2} \)
19 \( 1 - 0.921iT - 19T^{2} \)
23 \( 1 + 2.37T + 23T^{2} \)
29 \( 1 + 1.10iT - 29T^{2} \)
31 \( 1 - 3.57iT - 31T^{2} \)
41 \( 1 + 4.25T + 41T^{2} \)
43 \( 1 - 6.87T + 43T^{2} \)
47 \( 1 + 3.07iT - 47T^{2} \)
53 \( 1 + 3.55iT - 53T^{2} \)
59 \( 1 - 3.24iT - 59T^{2} \)
61 \( 1 - 2.85iT - 61T^{2} \)
67 \( 1 + 4.14iT - 67T^{2} \)
71 \( 1 + 10.9T + 71T^{2} \)
73 \( 1 - 2.97iT - 73T^{2} \)
79 \( 1 - 10.1iT - 79T^{2} \)
83 \( 1 + 9.43iT - 83T^{2} \)
89 \( 1 + 12.2iT - 89T^{2} \)
97 \( 1 + 10.5T + 97T^{2} \)
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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

−8.842624507168926936543542706548, −8.131464087567906951932535303169, −7.32633677128432605191019093848, −6.58714594359080498394299381198, −5.84291539875525921327838183222, −5.32139061030078777012327788616, −4.33390608540939649599478824586, −3.27106651805509032103687670130, −2.65115407333676948440790619570, −1.88957589917520764692840080789, 0.37765910414989057177576011926, 1.57914160230325232875431158629, 2.69141659370417544808888379645, 3.88637344876218212370628368674, 4.34647963416137288459915084599, 5.17774629407921897059150087260, 5.82887873595039616908781008313, 6.81001516191562036344445248788, 7.57899750132538844646626509589, 8.076374640872143827083040812562

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