Properties

Label 2-1216-1.1-c1-0-9
Degree $2$
Conductor $1216$
Sign $1$
Analytic cond. $9.70980$
Root an. cond. $3.11605$
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  − 1.56·3-s + 0.844·5-s + 5.06·7-s − 0.561·9-s + 2.84·11-s − 2.90·13-s − 1.31·15-s + 7.72·17-s + 19-s − 7.91·21-s − 3.91·23-s − 4.28·25-s + 5.56·27-s − 2.90·29-s + 7.00·31-s − 4.44·33-s + 4.27·35-s − 9.00·37-s + 4.53·39-s − 6.81·41-s + 5.96·43-s − 0.474·45-s + 1.04·47-s + 18.6·49-s − 12.0·51-s + 9.03·53-s + 2.40·55-s + ⋯
L(s)  = 1  − 0.901·3-s + 0.377·5-s + 1.91·7-s − 0.187·9-s + 0.857·11-s − 0.804·13-s − 0.340·15-s + 1.87·17-s + 0.229·19-s − 1.72·21-s − 0.815·23-s − 0.857·25-s + 1.07·27-s − 0.538·29-s + 1.25·31-s − 0.773·33-s + 0.723·35-s − 1.48·37-s + 0.725·39-s − 1.06·41-s + 0.910·43-s − 0.0707·45-s + 0.151·47-s + 2.66·49-s − 1.68·51-s + 1.24·53-s + 0.323·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $1$
Analytic conductor: \(9.70980\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.625805220\)
\(L(\frac12)\) \(\approx\) \(1.625805220\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 - T \)
good3 \( 1 + 1.56T + 3T^{2} \)
5 \( 1 - 0.844T + 5T^{2} \)
7 \( 1 - 5.06T + 7T^{2} \)
11 \( 1 - 2.84T + 11T^{2} \)
13 \( 1 + 2.90T + 13T^{2} \)
17 \( 1 - 7.72T + 17T^{2} \)
23 \( 1 + 3.91T + 23T^{2} \)
29 \( 1 + 2.90T + 29T^{2} \)
31 \( 1 - 7.00T + 31T^{2} \)
37 \( 1 + 9.00T + 37T^{2} \)
41 \( 1 + 6.81T + 41T^{2} \)
43 \( 1 - 5.96T + 43T^{2} \)
47 \( 1 - 1.04T + 47T^{2} \)
53 \( 1 - 9.03T + 53T^{2} \)
59 \( 1 - 0.749T + 59T^{2} \)
61 \( 1 + 2.27T + 61T^{2} \)
67 \( 1 - 10.3T + 67T^{2} \)
71 \( 1 + 2.13T + 71T^{2} \)
73 \( 1 - 4.60T + 73T^{2} \)
79 \( 1 + 3.88T + 79T^{2} \)
83 \( 1 - 8.44T + 83T^{2} \)
89 \( 1 - 16.0T + 89T^{2} \)
97 \( 1 + 3.68T + 97T^{2} \)
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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

−9.984717418486069826058117385089, −8.850747929723086901585969437800, −8.009574157247974528123517401790, −7.36587527251920445977263358886, −6.17197129387899924194818303283, −5.38934180538649206058523583571, −4.92573797800613736282758985059, −3.75250334344436521117111165906, −2.13620304908082963292739725866, −1.08078416561479399268482644582, 1.08078416561479399268482644582, 2.13620304908082963292739725866, 3.75250334344436521117111165906, 4.92573797800613736282758985059, 5.38934180538649206058523583571, 6.17197129387899924194818303283, 7.36587527251920445977263358886, 8.009574157247974528123517401790, 8.850747929723086901585969437800, 9.984717418486069826058117385089

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