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Properties

Label 2-3300-5.4-c1-0-25
Degree $2$
Conductor $3300$
Sign $0.447 + 0.894i$
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  + i·3-s + 5i·7-s − 9-s + 11-s − 4i·13-s − 5i·17-s − 7·19-s − 5·21-s − 9i·23-s − i·27-s − 2·29-s + 4·31-s + i·33-s − 7i·37-s + 4·39-s + ⋯
L(s)  = 1  + 0.577i·3-s + 1.88i·7-s − 0.333·9-s + 0.301·11-s − 1.10i·13-s − 1.21i·17-s − 1.60·19-s − 1.09·21-s − 1.87i·23-s − 0.192i·27-s − 0.371·29-s + 0.718·31-s + 0.174i·33-s − 1.15i·37-s + 0.640·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3300\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 11\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(26.3506\)
Root analytic conductor: \(5.13328\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3300} (1849, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3300,\ (\ :1/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9455614266\)
\(L(\frac12)\) \(\approx\) \(0.9455614266\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
5 \( 1 \)
11 \( 1 - T \)
good7 \( 1 - 5iT - 7T^{2} \)
13 \( 1 + 4iT - 13T^{2} \)
17 \( 1 + 5iT - 17T^{2} \)
19 \( 1 + 7T + 19T^{2} \)
23 \( 1 + 9iT - 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + 7iT - 37T^{2} \)
41 \( 1 + 7T + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 9iT - 47T^{2} \)
53 \( 1 + 2iT - 53T^{2} \)
59 \( 1 - 7T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 - 2iT - 67T^{2} \)
71 \( 1 - 5T + 71T^{2} \)
73 \( 1 - 2iT - 73T^{2} \)
79 \( 1 - 3T + 79T^{2} \)
83 \( 1 - 8iT - 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + 5iT - 97T^{2} \)
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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

−8.600734221920180781536285501124, −8.127608211633955870219127793616, −6.81280066809831837516043212899, −6.16569792283875429237264787717, −5.35346509963234639913381341537, −4.86050885664956734290855483223, −3.77875162988051987180092245934, −2.66202008511184909679061756834, −2.27427924458029120633079065951, −0.29071545410054502107150941104, 1.23485176004994209808944674088, 1.87747653622027353258055656279, 3.42917160002777061399864829890, 4.05944301464951918035624823986, 4.72215036738542870631513036192, 6.06217489361783577611618005843, 6.63474521964704277139092233716, 7.19447473080166777717534365303, 7.956926097701186637737085357111, 8.572914128764437921643817179211

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