Properties

Label 2-3696-1.1-c1-0-25
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 + 2.39·5-s + 7-s + 9-s + 11-s + 5.04·13-s − 2.39·15-s − 6.36·17-s + 5.32·19-s − 21-s − 4.92·23-s + 0.751·25-s − 27-s + 5.04·29-s + 7.57·31-s − 33-s + 2.39·35-s + 4.24·37-s − 5.04·39-s − 0.646·41-s + 10.5·43-s + 2.39·45-s − 0.526·47-s + 49-s + 6.36·51-s + 3.72·53-s + 2.39·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.07·5-s + 0.377·7-s + 0.333·9-s + 0.301·11-s + 1.39·13-s − 0.619·15-s − 1.54·17-s + 1.22·19-s − 0.218·21-s − 1.02·23-s + 0.150·25-s − 0.192·27-s + 0.936·29-s + 1.35·31-s − 0.174·33-s + 0.405·35-s + 0.698·37-s − 0.807·39-s − 0.101·41-s + 1.60·43-s + 0.357·45-s − 0.0768·47-s + 0.142·49-s + 0.891·51-s + 0.511·53-s + 0.323·55-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

\(L(1)\) \(\approx\) \(2.251652089\)
\(L(\frac12)\) \(\approx\) \(2.251652089\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.39T + 5T^{2} \)
13 \( 1 - 5.04T + 13T^{2} \)
17 \( 1 + 6.36T + 17T^{2} \)
19 \( 1 - 5.32T + 19T^{2} \)
23 \( 1 + 4.92T + 23T^{2} \)
29 \( 1 - 5.04T + 29T^{2} \)
31 \( 1 - 7.57T + 31T^{2} \)
37 \( 1 - 4.24T + 37T^{2} \)
41 \( 1 + 0.646T + 41T^{2} \)
43 \( 1 - 10.5T + 43T^{2} \)
47 \( 1 + 0.526T + 47T^{2} \)
53 \( 1 - 3.72T + 53T^{2} \)
59 \( 1 + 7.97T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 8.76T + 67T^{2} \)
71 \( 1 - 11.4T + 71T^{2} \)
73 \( 1 + 13.0T + 73T^{2} \)
79 \( 1 + 11.4T + 79T^{2} \)
83 \( 1 + 13.1T + 83T^{2} \)
89 \( 1 - 11.8T + 89T^{2} \)
97 \( 1 + 1.87T + 97T^{2} \)
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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

−8.648844942867687703545943298357, −7.79558109172219392353656992085, −6.83660427491770694418893151981, −6.01486432050076410452735179307, −5.90135329968953857902734340282, −4.70596156251043827502147211548, −4.12751797562809132546973901229, −2.86061801858271483756255677168, −1.84120696627217694691777653612, −0.964760919634283298860973572278, 0.964760919634283298860973572278, 1.84120696627217694691777653612, 2.86061801858271483756255677168, 4.12751797562809132546973901229, 4.70596156251043827502147211548, 5.90135329968953857902734340282, 6.01486432050076410452735179307, 6.83660427491770694418893151981, 7.79558109172219392353656992085, 8.648844942867687703545943298357

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