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Properties

Label 2-3300-55.32-c1-0-32
Degree $2$
Conductor $3300$
Sign $-0.315 + 0.948i$
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 + (3.37 − 3.37i)7-s − 1.00i·9-s + (−3.25 − 0.626i)11-s + (1.72 + 1.72i)13-s + (0.580 − 0.580i)17-s − 4.50·19-s − 4.77i·21-s + (3.65 − 3.65i)23-s + (−0.707 − 0.707i)27-s + 0.725·29-s + 6.16·31-s + (−2.74 + 1.86i)33-s + (0.978 + 0.978i)37-s + 2.43·39-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (1.27 − 1.27i)7-s − 0.333i·9-s + (−0.982 − 0.188i)11-s + (0.477 + 0.477i)13-s + (0.140 − 0.140i)17-s − 1.03·19-s − 1.04i·21-s + (0.761 − 0.761i)23-s + (−0.136 − 0.136i)27-s + 0.134·29-s + 1.10·31-s + (−0.477 + 0.323i)33-s + (0.160 + 0.160i)37-s + 0.390·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.315 + 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.315 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3300\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 11\)
Sign: $-0.315 + 0.948i$
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: \((2,\ 3300,\ (\ :1/2),\ -0.315 + 0.948i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.190533538\)
\(L(\frac12)\) \(\approx\) \(2.190533538\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.25 + 0.626i)T \)
good7 \( 1 + (-3.37 + 3.37i)T - 7iT^{2} \)
13 \( 1 + (-1.72 - 1.72i)T + 13iT^{2} \)
17 \( 1 + (-0.580 + 0.580i)T - 17iT^{2} \)
19 \( 1 + 4.50T + 19T^{2} \)
23 \( 1 + (-3.65 + 3.65i)T - 23iT^{2} \)
29 \( 1 - 0.725T + 29T^{2} \)
31 \( 1 - 6.16T + 31T^{2} \)
37 \( 1 + (-0.978 - 0.978i)T + 37iT^{2} \)
41 \( 1 + 8.99iT - 41T^{2} \)
43 \( 1 + (-0.936 - 0.936i)T + 43iT^{2} \)
47 \( 1 + (8.25 + 8.25i)T + 47iT^{2} \)
53 \( 1 + (0.742 - 0.742i)T - 53iT^{2} \)
59 \( 1 + 9.89iT - 59T^{2} \)
61 \( 1 - 14.4iT - 61T^{2} \)
67 \( 1 + (1.66 + 1.66i)T + 67iT^{2} \)
71 \( 1 - 4.84T + 71T^{2} \)
73 \( 1 + (-8.52 - 8.52i)T + 73iT^{2} \)
79 \( 1 + 8.00T + 79T^{2} \)
83 \( 1 + (8.89 + 8.89i)T + 83iT^{2} \)
89 \( 1 + 12.7iT - 89T^{2} \)
97 \( 1 + (1.29 + 1.29i)T + 97iT^{2} \)
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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

−8.408521314860562640939528049053, −7.71271589688001183891059668479, −7.05617837437956464457091578626, −6.36332919427928653619910768683, −5.19359742118743753661592406015, −4.52184620363733712789410538760, −3.77343779301106092944732902822, −2.62331259348746764099643842825, −1.68266045584175801443959965857, −0.63218683115640202015513868136, 1.46775741956784080573290421953, 2.45514355700487180551069693202, 3.11470350224142173604964896121, 4.41664866120266358864814483536, 5.00537463837637992227574622514, 5.65731873820794547841572358757, 6.50693865964880509958726317613, 7.87346162278694477954416105762, 8.049524200648732542638603563975, 8.732853035704359779454095428119

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