Properties

Label 2-3696-1.1-c1-0-29
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 + 3·5-s − 7-s + 9-s + 11-s + 13-s + 3·15-s − 1.58·17-s − 2.58·19-s − 21-s − 3.58·23-s + 4·25-s + 27-s + 10.1·29-s + 5.58·31-s + 33-s − 3·35-s + 37-s + 39-s + 7.16·41-s + 7.58·43-s + 3·45-s − 10.5·47-s + 49-s − 1.58·51-s − 0.417·53-s + 3·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.34·5-s − 0.377·7-s + 0.333·9-s + 0.301·11-s + 0.277·13-s + 0.774·15-s − 0.383·17-s − 0.592·19-s − 0.218·21-s − 0.747·23-s + 0.800·25-s + 0.192·27-s + 1.88·29-s + 1.00·31-s + 0.174·33-s − 0.507·35-s + 0.164·37-s + 0.160·39-s + 1.11·41-s + 1.15·43-s + 0.447·45-s − 1.54·47-s + 0.142·49-s − 0.221·51-s − 0.0573·53-s + 0.404·55-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

\(L(1)\) \(\approx\) \(3.116720402\)
\(L(\frac12)\) \(\approx\) \(3.116720402\)
\(L(\frac{3}{2})\) 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 \( 1 - 3T + 5T^{2} \)
13 \( 1 - T + 13T^{2} \)
17 \( 1 + 1.58T + 17T^{2} \)
19 \( 1 + 2.58T + 19T^{2} \)
23 \( 1 + 3.58T + 23T^{2} \)
29 \( 1 - 10.1T + 29T^{2} \)
31 \( 1 - 5.58T + 31T^{2} \)
37 \( 1 - T + 37T^{2} \)
41 \( 1 - 7.16T + 41T^{2} \)
43 \( 1 - 7.58T + 43T^{2} \)
47 \( 1 + 10.5T + 47T^{2} \)
53 \( 1 + 0.417T + 53T^{2} \)
59 \( 1 - 4.58T + 59T^{2} \)
61 \( 1 - 10T + 61T^{2} \)
67 \( 1 - 0.582T + 67T^{2} \)
71 \( 1 - 7.16T + 71T^{2} \)
73 \( 1 - 7T + 73T^{2} \)
79 \( 1 - 11.1T + 79T^{2} \)
83 \( 1 - 2.41T + 83T^{2} \)
89 \( 1 + 9.16T + 89T^{2} \)
97 \( 1 + 11.5T + 97T^{2} \)
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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

−8.500689446833055001099449387147, −8.042568480195725916233526167866, −6.74534472074773064211733407631, −6.44149624567824050826581296683, −5.66992594967210592452856750986, −4.67532184049123285660327163647, −3.87537894885409196915233245817, −2.73441290422602185310196572047, −2.18253588002634172535927680470, −1.05219458621003520936746611663, 1.05219458621003520936746611663, 2.18253588002634172535927680470, 2.73441290422602185310196572047, 3.87537894885409196915233245817, 4.67532184049123285660327163647, 5.66992594967210592452856750986, 6.44149624567824050826581296683, 6.74534472074773064211733407631, 8.042568480195725916233526167866, 8.500689446833055001099449387147

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