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Properties

Label 2-3300-55.32-c1-0-17
Degree $2$
Conductor $3300$
Sign $0.990 - 0.140i$
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 + (0.256 − 0.256i)7-s − 1.00i·9-s + (0.890 + 3.19i)11-s + (0.580 + 0.580i)13-s + (1.72 − 1.72i)17-s + 3.13·19-s − 0.362i·21-s + (−4.36 + 4.36i)23-s + (−0.707 − 0.707i)27-s + 9.54·29-s − 5.16·31-s + (2.88 + 1.62i)33-s + (−1.16 − 1.16i)37-s + 0.821·39-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (0.0969 − 0.0969i)7-s − 0.333i·9-s + (0.268 + 0.963i)11-s + (0.161 + 0.161i)13-s + (0.417 − 0.417i)17-s + 0.718·19-s − 0.0791i·21-s + (−0.909 + 0.909i)23-s + (−0.136 − 0.136i)27-s + 1.77·29-s − 0.927·31-s + (0.502 + 0.283i)33-s + (−0.192 − 0.192i)37-s + 0.131·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.140i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 - 0.140i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3300\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 11\)
Sign: $0.990 - 0.140i$
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: \((2,\ 3300,\ (\ :1/2),\ 0.990 - 0.140i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.275331329\)
\(L(\frac12)\) \(\approx\) \(2.275331329\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.890 - 3.19i)T \)
good7 \( 1 + (-0.256 + 0.256i)T - 7iT^{2} \)
13 \( 1 + (-0.580 - 0.580i)T + 13iT^{2} \)
17 \( 1 + (-1.72 + 1.72i)T - 17iT^{2} \)
19 \( 1 - 3.13T + 19T^{2} \)
23 \( 1 + (4.36 - 4.36i)T - 23iT^{2} \)
29 \( 1 - 9.54T + 29T^{2} \)
31 \( 1 + 5.16T + 31T^{2} \)
37 \( 1 + (1.16 + 1.16i)T + 37iT^{2} \)
41 \( 1 + 1.78iT - 41T^{2} \)
43 \( 1 + (-2.07 - 2.07i)T + 43iT^{2} \)
47 \( 1 + (-5.62 - 5.62i)T + 47iT^{2} \)
53 \( 1 + (-5.12 + 5.12i)T - 53iT^{2} \)
59 \( 1 - 1.43iT - 59T^{2} \)
61 \( 1 - 11.6iT - 61T^{2} \)
67 \( 1 + (3.80 + 3.80i)T + 67iT^{2} \)
71 \( 1 - 1.81T + 71T^{2} \)
73 \( 1 + (8.52 + 8.52i)T + 73iT^{2} \)
79 \( 1 - 11.2T + 79T^{2} \)
83 \( 1 + (-3.57 - 3.57i)T + 83iT^{2} \)
89 \( 1 - 1.62iT - 89T^{2} \)
97 \( 1 + (-10.4 - 10.4i)T + 97iT^{2} \)
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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

−8.696438630290792658273094371085, −7.61243281584488710681293814545, −7.45744568277807067252451878095, −6.49345123189531109755788015787, −5.69294581902318273652763858234, −4.75425880025047239123705393708, −3.93141649195529933797294000736, −3.00814362553216410740403727442, −2.02090774039371666811069807018, −1.05474881078796556443174449580, 0.797921497151537257962277502158, 2.12932244976692532769701519833, 3.15928840692937987112919354096, 3.79882659162706177564468538303, 4.73181562811205263876441648826, 5.61291358002711057973814633119, 6.26765155861481543854748648045, 7.19981909081065170826496846846, 8.126040181782419845073068109397, 8.556574921799537123559772183853

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