Properties

Label 2-44e2-1.1-c3-0-117
Degree $2$
Conductor $1936$
Sign $-1$
Analytic cond. $114.227$
Root an. cond. $10.6877$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7.57·3-s + 5.40·5-s + 22.1·7-s + 30.3·9-s + 76.8·13-s − 40.9·15-s + 59.2·17-s − 95.2·19-s − 167.·21-s − 142.·23-s − 95.7·25-s − 25.6·27-s − 20.4·29-s − 213.·31-s + 119.·35-s − 145.·37-s − 582.·39-s + 82.8·41-s + 151.·43-s + 164.·45-s − 90.3·47-s + 147.·49-s − 449.·51-s − 234.·53-s + 721.·57-s − 302.·59-s − 149.·61-s + ⋯
L(s)  = 1  − 1.45·3-s + 0.483·5-s + 1.19·7-s + 1.12·9-s + 1.63·13-s − 0.705·15-s + 0.845·17-s − 1.15·19-s − 1.74·21-s − 1.29·23-s − 0.766·25-s − 0.183·27-s − 0.130·29-s − 1.23·31-s + 0.578·35-s − 0.646·37-s − 2.38·39-s + 0.315·41-s + 0.536·43-s + 0.544·45-s − 0.280·47-s + 0.430·49-s − 1.23·51-s − 0.608·53-s + 1.67·57-s − 0.667·59-s − 0.313·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1936 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1936\)    =    \(2^{4} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(114.227\)
Root analytic conductor: \(10.6877\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1936,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
good3 \( 1 + 7.57T + 27T^{2} \)
5 \( 1 - 5.40T + 125T^{2} \)
7 \( 1 - 22.1T + 343T^{2} \)
13 \( 1 - 76.8T + 2.19e3T^{2} \)
17 \( 1 - 59.2T + 4.91e3T^{2} \)
19 \( 1 + 95.2T + 6.85e3T^{2} \)
23 \( 1 + 142.T + 1.21e4T^{2} \)
29 \( 1 + 20.4T + 2.43e4T^{2} \)
31 \( 1 + 213.T + 2.97e4T^{2} \)
37 \( 1 + 145.T + 5.06e4T^{2} \)
41 \( 1 - 82.8T + 6.89e4T^{2} \)
43 \( 1 - 151.T + 7.95e4T^{2} \)
47 \( 1 + 90.3T + 1.03e5T^{2} \)
53 \( 1 + 234.T + 1.48e5T^{2} \)
59 \( 1 + 302.T + 2.05e5T^{2} \)
61 \( 1 + 149.T + 2.26e5T^{2} \)
67 \( 1 + 826.T + 3.00e5T^{2} \)
71 \( 1 - 898.T + 3.57e5T^{2} \)
73 \( 1 + 137.T + 3.89e5T^{2} \)
79 \( 1 - 304.T + 4.93e5T^{2} \)
83 \( 1 - 764.T + 5.71e5T^{2} \)
89 \( 1 + 313.T + 7.04e5T^{2} \)
97 \( 1 + 582.T + 9.12e5T^{2} \)
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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

−8.330675556093216504647089880377, −7.70988687114217263747472618166, −6.53134211722120207777453783178, −5.91277781717168167058725334013, −5.46791064519424492749877204619, −4.50115493962109934537005616677, −3.69208106538011908116223916334, −1.93204854835355817059576715164, −1.27636711695212601717034269464, 0, 1.27636711695212601717034269464, 1.93204854835355817059576715164, 3.69208106538011908116223916334, 4.50115493962109934537005616677, 5.46791064519424492749877204619, 5.91277781717168167058725334013, 6.53134211722120207777453783178, 7.70988687114217263747472618166, 8.330675556093216504647089880377

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