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Properties

Label 2-3330-5.4-c1-0-45
Degree $2$
Conductor $3330$
Sign $0.929 + 0.369i$
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 + (−2.07 − 0.826i)5-s − 4.67i·7-s − i·8-s + (0.826 − 2.07i)10-s − 0.0451·11-s + 5.26i·13-s + 4.67·14-s + 16-s + 3.60i·17-s + 6.22·19-s + (2.07 + 0.826i)20-s − 0.0451i·22-s + 2.20i·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + (−0.929 − 0.369i)5-s − 1.76i·7-s − 0.353i·8-s + (0.261 − 0.656i)10-s − 0.0136·11-s + 1.46i·13-s + 1.24·14-s + 0.250·16-s + 0.875i·17-s + 1.42·19-s + (0.464 + 0.184i)20-s − 0.00962i·22-s + 0.459i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 + 0.369i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.929 + 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3330\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 37\)
Sign: $0.929 + 0.369i$
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: \((2,\ 3330,\ (\ :1/2),\ 0.929 + 0.369i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.242385349\)
\(L(\frac12)\) \(\approx\) \(1.242385349\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.07 + 0.826i)T \)
37 \( 1 + iT \)
good7 \( 1 + 4.67iT - 7T^{2} \)
11 \( 1 + 0.0451T + 11T^{2} \)
13 \( 1 - 5.26iT - 13T^{2} \)
17 \( 1 - 3.60iT - 17T^{2} \)
19 \( 1 - 6.22T + 19T^{2} \)
23 \( 1 - 2.20iT - 23T^{2} \)
29 \( 1 + 4.20T + 29T^{2} \)
31 \( 1 + 3.01T + 31T^{2} \)
41 \( 1 - 7.38T + 41T^{2} \)
43 \( 1 + 5.54iT - 43T^{2} \)
47 \( 1 + 4.28iT - 47T^{2} \)
53 \( 1 + 6.10iT - 53T^{2} \)
59 \( 1 - 13.5T + 59T^{2} \)
61 \( 1 + 4.27T + 61T^{2} \)
67 \( 1 + 12.3iT - 67T^{2} \)
71 \( 1 - 4.49T + 71T^{2} \)
73 \( 1 - 7.03iT - 73T^{2} \)
79 \( 1 - 8.72T + 79T^{2} \)
83 \( 1 - 0.880iT - 83T^{2} \)
89 \( 1 - 9.97T + 89T^{2} \)
97 \( 1 + 0.240iT - 97T^{2} \)
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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

−8.379327814305137717946749053790, −7.61794498378735648076081707490, −7.22041709880022152244581660733, −6.64988435139270175694502076849, −5.49789391307077471892364960446, −4.65803506329269988787531628205, −3.84865184716299926590080838945, −3.63398472270167328905437470923, −1.62344997431544734599408318251, −0.53940026411688828613713336718, 0.859819334857564487901276513036, 2.47783655816558234561235745211, 2.91429992809403701656967970517, 3.70414797217783657629331417126, 4.95614572021263443641393898013, 5.43043911396622875897828813254, 6.26494207915933409499450137949, 7.55078728251489641186206074181, 7.87742708157636472854583974888, 8.841400960309479982937635207151

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