Properties

Label 2-3330-5.4-c1-0-35
Degree $2$
Conductor $3330$
Sign $0.762 - 0.646i$
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  + i·2-s − 4-s + (−1.70 + 1.44i)5-s + 1.83i·7-s i·8-s + (−1.44 − 1.70i)10-s − 4.19·11-s − 0.369i·13-s − 1.83·14-s + 16-s − 5.08i·17-s − 3.55·19-s + (1.70 − 1.44i)20-s − 4.19i·22-s − 5.62i·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + (−0.762 + 0.646i)5-s + 0.692i·7-s − 0.353i·8-s + (−0.457 − 0.539i)10-s − 1.26·11-s − 0.102i·13-s − 0.489·14-s + 0.250·16-s − 1.23i·17-s − 0.816·19-s + (0.381 − 0.323i)20-s − 0.893i·22-s − 1.17i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.762 - 0.646i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.762 - 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3330\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 37\)
Sign: $0.762 - 0.646i$
Analytic conductor: \(26.5901\)
Root analytic conductor: \(5.15656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3330} (1999, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3330,\ (\ :1/2),\ 0.762 - 0.646i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.006580505\)
\(L(\frac12)\) \(\approx\) \(1.006580505\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 \)
5 \( 1 + (1.70 - 1.44i)T \)
37 \( 1 + iT \)
good7 \( 1 - 1.83iT - 7T^{2} \)
11 \( 1 + 4.19T + 11T^{2} \)
13 \( 1 + 0.369iT - 13T^{2} \)
17 \( 1 + 5.08iT - 17T^{2} \)
19 \( 1 + 3.55T + 19T^{2} \)
23 \( 1 + 5.62iT - 23T^{2} \)
29 \( 1 - 1.20T + 29T^{2} \)
31 \( 1 - 10.1T + 31T^{2} \)
41 \( 1 - 8.01T + 41T^{2} \)
43 \( 1 + 2.27iT - 43T^{2} \)
47 \( 1 - 10.9iT - 47T^{2} \)
53 \( 1 - 9.94iT - 53T^{2} \)
59 \( 1 + 5.34T + 59T^{2} \)
61 \( 1 + 9.79T + 61T^{2} \)
67 \( 1 + 1.85iT - 67T^{2} \)
71 \( 1 + 2.86T + 71T^{2} \)
73 \( 1 + 8.09iT - 73T^{2} \)
79 \( 1 + 6.06T + 79T^{2} \)
83 \( 1 - 8.93iT - 83T^{2} \)
89 \( 1 - 11.4T + 89T^{2} \)
97 \( 1 + 6.05iT - 97T^{2} \)
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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

−8.541550114608952158217618413329, −7.81745865973556363886708518103, −7.39250689840680224470854924767, −6.40870575666562373713227886572, −5.90761900483548697577990984738, −4.77063895592639332855835644930, −4.39165558163480169956839149619, −2.90492108718451514288979709820, −2.59153088052073217173869008683, −0.48394688148472140484538998146, 0.70630505344138837723702164522, 1.84167244400011722144110523098, 3.00137480810940827372194410762, 3.90242772226492802888661123398, 4.48254542860458472325200169098, 5.26950054006794484675485138465, 6.20468350030554791783228306324, 7.30787130248792377263203063317, 8.022567297833060200138682427579, 8.411551199362282342870930056932

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