Properties

Label 2-3300-5.4-c1-0-0
Degree $2$
Conductor $3300$
Sign $-0.894 - 0.447i$
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 − 2.60i·7-s − 9-s + 11-s − 2.60i·13-s − 0.605i·17-s − 7.21·19-s + 2.60·21-s i·27-s − 8·29-s − 5.21·31-s + i·33-s + 11.2i·37-s + 2.60·39-s + 8·41-s + ⋯
L(s)  = 1  + 0.577i·3-s − 0.984i·7-s − 0.333·9-s + 0.301·11-s − 0.722i·13-s − 0.146i·17-s − 1.65·19-s + 0.568·21-s − 0.192i·27-s − 1.48·29-s − 0.935·31-s + 0.174i·33-s + 1.84i·37-s + 0.417·39-s + 1.24·41-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3300\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 11\)
Sign: $-0.894 - 0.447i$
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: \((2,\ 3300,\ (\ :1/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4222461048\)
\(L(\frac12)\) \(\approx\) \(0.4222461048\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.60iT - 7T^{2} \)
13 \( 1 + 2.60iT - 13T^{2} \)
17 \( 1 + 0.605iT - 17T^{2} \)
19 \( 1 + 7.21T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 8T + 29T^{2} \)
31 \( 1 + 5.21T + 31T^{2} \)
37 \( 1 - 11.2iT - 37T^{2} \)
41 \( 1 - 8T + 41T^{2} \)
43 \( 1 - 10.6iT - 43T^{2} \)
47 \( 1 - 9.21iT - 47T^{2} \)
53 \( 1 + 2iT - 53T^{2} \)
59 \( 1 + 8T + 59T^{2} \)
61 \( 1 + 7.21T + 61T^{2} \)
67 \( 1 + 4iT - 67T^{2} \)
71 \( 1 - 14.4T + 71T^{2} \)
73 \( 1 - 6.60iT - 73T^{2} \)
79 \( 1 + 3.21T + 79T^{2} \)
83 \( 1 - 3.39iT - 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 - 16.4iT - 97T^{2} \)
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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

−9.109032783079850494812679827559, −8.087556880091551882909737985885, −7.64148599166016993430737632672, −6.61764519302491371338797429085, −6.00968290049463293342213999744, −4.98749970466357472797445515224, −4.25502956561474824754306467053, −3.62039762142240966941006398415, −2.61254099499821002032468131932, −1.29045020722135178632285407707, 0.12509062827293365449287574440, 1.90445315514832568149101720024, 2.23641427897853549912387759932, 3.60993545237907374867181729308, 4.35987577202557137507589290502, 5.59751076781682402687921233825, 5.93133206076469363370762795269, 6.90431537862308458903484983250, 7.45134604349487453094999917468, 8.476148134269736349492985247867

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