Properties

Label 2-3696-1.1-c1-0-28
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 + 4.18·5-s + 7-s + 9-s + 11-s − 3.17·13-s − 4.18·15-s + 6.85·17-s + 0.318·19-s − 21-s + 1.87·23-s + 12.5·25-s − 27-s − 3.17·29-s − 9.23·31-s − 33-s + 4.18·35-s − 7.55·37-s + 3.17·39-s + 9.36·41-s + 10.8·43-s + 4.18·45-s + 8.06·47-s + 49-s − 6.85·51-s + 0.508·53-s + 4.18·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.87·5-s + 0.377·7-s + 0.333·9-s + 0.301·11-s − 0.880·13-s − 1.08·15-s + 1.66·17-s + 0.0731·19-s − 0.218·21-s + 0.390·23-s + 2.51·25-s − 0.192·27-s − 0.589·29-s − 1.65·31-s − 0.174·33-s + 0.708·35-s − 1.24·37-s + 0.508·39-s + 1.46·41-s + 1.66·43-s + 0.624·45-s + 1.17·47-s + 0.142·49-s − 0.959·51-s + 0.0698·53-s + 0.564·55-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3696,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.551942763\)
\(L(\frac12)\) \(\approx\) \(2.551942763\)
\(L(\frac{3}{2})\) 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 \( 1 - 4.18T + 5T^{2} \)
13 \( 1 + 3.17T + 13T^{2} \)
17 \( 1 - 6.85T + 17T^{2} \)
19 \( 1 - 0.318T + 19T^{2} \)
23 \( 1 - 1.87T + 23T^{2} \)
29 \( 1 + 3.17T + 29T^{2} \)
31 \( 1 + 9.23T + 31T^{2} \)
37 \( 1 + 7.55T + 37T^{2} \)
41 \( 1 - 9.36T + 41T^{2} \)
43 \( 1 - 10.8T + 43T^{2} \)
47 \( 1 - 8.06T + 47T^{2} \)
53 \( 1 - 0.508T + 53T^{2} \)
59 \( 1 - 7.04T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 - 2.66T + 67T^{2} \)
71 \( 1 - 5.01T + 71T^{2} \)
73 \( 1 + 4.82T + 73T^{2} \)
79 \( 1 + 5.01T + 79T^{2} \)
83 \( 1 + 3.52T + 83T^{2} \)
89 \( 1 + 1.74T + 89T^{2} \)
97 \( 1 + 12.2T + 97T^{2} \)
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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

−8.765666795221537768398354932273, −7.49788177438515530215528087373, −7.09734673673948333306587917000, −6.01932050886035694863227706430, −5.52194199390554677415861790925, −5.15921322296804708298016368780, −3.97929745791476547864284670174, −2.75868645761216958515863969917, −1.89182752987041465621278356610, −1.03220033152491000292246679950, 1.03220033152491000292246679950, 1.89182752987041465621278356610, 2.75868645761216958515863969917, 3.97929745791476547864284670174, 5.15921322296804708298016368780, 5.52194199390554677415861790925, 6.01932050886035694863227706430, 7.09734673673948333306587917000, 7.49788177438515530215528087373, 8.765666795221537768398354932273

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