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Properties

Label 2-3300-55.32-c1-0-1
Degree $2$
Conductor $3300$
Sign $-0.735 + 0.677i$
Analytic cond. $26.3506$
Root an. cond. $5.13328$
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  + (−0.707 + 0.707i)3-s + (−2.08 + 2.08i)7-s − 1.00i·9-s + (2.54 + 2.12i)11-s + (−0.270 − 0.270i)13-s + (−3.69 + 3.69i)17-s − 1.35·19-s − 2.95i·21-s + (−1.12 + 1.12i)23-s + (0.707 + 0.707i)27-s − 1.17·29-s + 2.59·31-s + (−3.30 + 0.297i)33-s + (−4.21 − 4.21i)37-s + 0.383·39-s + ⋯
L(s)  = 1  + (−0.408 + 0.408i)3-s + (−0.789 + 0.789i)7-s − 0.333i·9-s + (0.767 + 0.640i)11-s + (−0.0751 − 0.0751i)13-s + (−0.895 + 0.895i)17-s − 0.310·19-s − 0.644i·21-s + (−0.235 + 0.235i)23-s + (0.136 + 0.136i)27-s − 0.217·29-s + 0.466·31-s + (−0.575 + 0.0518i)33-s + (−0.692 − 0.692i)37-s + 0.0613·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.735 + 0.677i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.735 + 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3300\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 11\)
Sign: $-0.735 + 0.677i$
Analytic conductor: \(26.3506\)
Root analytic conductor: \(5.13328\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3300} (1957, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3300,\ (\ :1/2),\ -0.735 + 0.677i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1996097070\)
\(L(\frac12)\) \(\approx\) \(0.1996097070\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 \)
11 \( 1 + (-2.54 - 2.12i)T \)
good7 \( 1 + (2.08 - 2.08i)T - 7iT^{2} \)
13 \( 1 + (0.270 + 0.270i)T + 13iT^{2} \)
17 \( 1 + (3.69 - 3.69i)T - 17iT^{2} \)
19 \( 1 + 1.35T + 19T^{2} \)
23 \( 1 + (1.12 - 1.12i)T - 23iT^{2} \)
29 \( 1 + 1.17T + 29T^{2} \)
31 \( 1 - 2.59T + 31T^{2} \)
37 \( 1 + (4.21 + 4.21i)T + 37iT^{2} \)
41 \( 1 + 2.29iT - 41T^{2} \)
43 \( 1 + (-4.84 - 4.84i)T + 43iT^{2} \)
47 \( 1 + (2.47 + 2.47i)T + 47iT^{2} \)
53 \( 1 + (2.56 - 2.56i)T - 53iT^{2} \)
59 \( 1 + 2.86iT - 59T^{2} \)
61 \( 1 - 4.07iT - 61T^{2} \)
67 \( 1 + (4.02 + 4.02i)T + 67iT^{2} \)
71 \( 1 - 2.49T + 71T^{2} \)
73 \( 1 + (-1.83 - 1.83i)T + 73iT^{2} \)
79 \( 1 + 1.86T + 79T^{2} \)
83 \( 1 + (-0.432 - 0.432i)T + 83iT^{2} \)
89 \( 1 + 3.35iT - 89T^{2} \)
97 \( 1 + (7.77 + 7.77i)T + 97iT^{2} \)
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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

−9.127158089871654811731558826741, −8.601377113164724058451853728785, −7.53880913749541133194589148837, −6.59398658111522904297332288745, −6.19087370557353309625505318177, −5.37496122152321630558354160113, −4.39627081585841878862394035873, −3.77076798764812639932919539057, −2.68549922749504405062383211355, −1.66559838256349131174571504960, 0.06962956277206555309747993840, 1.11941443365872416877876967176, 2.42203121082627325917193323696, 3.45283063766826871249994215232, 4.24966570462918046044939749388, 5.12389840870131232437117525956, 6.18742662799726822109971955403, 6.64464221995293766652552183314, 7.21494745026026583983127828950, 8.138929757930305978440790327574

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