Properties

Label 2-3696-1.1-c1-0-5
Degree $2$
Conductor $3696$
Sign $1$
Analytic cond. $29.5127$
Root an. cond. $5.43256$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2.58·5-s + 7-s + 9-s + 11-s − 5.87·13-s + 2.58·15-s + 7.51·17-s + 2.35·19-s − 21-s − 6.94·23-s + 1.69·25-s − 27-s − 5.87·29-s + 3.66·31-s − 33-s − 2.58·35-s + 3.30·37-s + 5.87·39-s + 5.28·41-s − 7.40·43-s − 2.58·45-s − 7.53·47-s + 49-s − 7.51·51-s − 4.22·53-s − 2.58·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.15·5-s + 0.377·7-s + 0.333·9-s + 0.301·11-s − 1.62·13-s + 0.668·15-s + 1.82·17-s + 0.540·19-s − 0.218·21-s − 1.44·23-s + 0.339·25-s − 0.192·27-s − 1.09·29-s + 0.657·31-s − 0.174·33-s − 0.437·35-s + 0.543·37-s + 0.940·39-s + 0.825·41-s − 1.12·43-s − 0.385·45-s − 1.09·47-s + 0.142·49-s − 1.05·51-s − 0.581·53-s − 0.348·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3696\)    =    \(2^{4} \cdot 3 \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(29.5127\)
Root analytic conductor: \(5.43256\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3696} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3696,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9327609283\)
\(L(\frac12)\) \(\approx\) \(0.9327609283\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
7 \( 1 - T \)
11 \( 1 - T \)
good5 \( 1 + 2.58T + 5T^{2} \)
13 \( 1 + 5.87T + 13T^{2} \)
17 \( 1 - 7.51T + 17T^{2} \)
19 \( 1 - 2.35T + 19T^{2} \)
23 \( 1 + 6.94T + 23T^{2} \)
29 \( 1 + 5.87T + 29T^{2} \)
31 \( 1 - 3.66T + 31T^{2} \)
37 \( 1 - 3.30T + 37T^{2} \)
41 \( 1 - 5.28T + 41T^{2} \)
43 \( 1 + 7.40T + 43T^{2} \)
47 \( 1 + 7.53T + 47T^{2} \)
53 \( 1 + 4.22T + 53T^{2} \)
59 \( 1 - 0.926T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 - 10.1T + 67T^{2} \)
71 \( 1 + 4.45T + 71T^{2} \)
73 \( 1 + 2.12T + 73T^{2} \)
79 \( 1 - 4.45T + 79T^{2} \)
83 \( 1 - 10.6T + 83T^{2} \)
89 \( 1 - 15.8T + 89T^{2} \)
97 \( 1 - 10.1T + 97T^{2} \)
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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

−8.129975850189929748169039310735, −7.77609034360705403553028278420, −7.29532204584028897990879339315, −6.27705042480643582005394971925, −5.40726541664698431168493589470, −4.77399972498254181123663549150, −3.94787196484250245389805423671, −3.17843405199396734062401179600, −1.88171481368921067785699520471, −0.57475897874472152869261985363, 0.57475897874472152869261985363, 1.88171481368921067785699520471, 3.17843405199396734062401179600, 3.94787196484250245389805423671, 4.77399972498254181123663549150, 5.40726541664698431168493589470, 6.27705042480643582005394971925, 7.29532204584028897990879339315, 7.77609034360705403553028278420, 8.129975850189929748169039310735

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