Properties

Label 2-1216-1.1-c1-0-6
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  + 0.786·3-s − 3.29·5-s + 2.08·7-s − 2.38·9-s + 1.29·11-s − 1.21·13-s − 2.59·15-s + 4.08·17-s − 19-s + 1.63·21-s + 8.95·23-s + 5.87·25-s − 4.23·27-s + 9.38·29-s + 1.02·33-s − 6.87·35-s + 2·37-s − 0.954·39-s + 3.57·41-s + 7.72·43-s + 7.85·45-s − 9.46·47-s − 2.65·49-s + 3.21·51-s + 11.9·53-s − 4.27·55-s − 0.786·57-s + ⋯
L(s)  = 1  + 0.454·3-s − 1.47·5-s + 0.787·7-s − 0.793·9-s + 0.391·11-s − 0.336·13-s − 0.669·15-s + 0.990·17-s − 0.229·19-s + 0.357·21-s + 1.86·23-s + 1.17·25-s − 0.814·27-s + 1.74·29-s + 0.177·33-s − 1.16·35-s + 0.328·37-s − 0.152·39-s + 0.558·41-s + 1.17·43-s + 1.17·45-s − 1.38·47-s − 0.379·49-s + 0.449·51-s + 1.64·53-s − 0.576·55-s − 0.104·57-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

\(L(1)\) \(\approx\) \(1.536559885\)
\(L(\frac12)\) \(\approx\) \(1.536559885\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.786T + 3T^{2} \)
5 \( 1 + 3.29T + 5T^{2} \)
7 \( 1 - 2.08T + 7T^{2} \)
11 \( 1 - 1.29T + 11T^{2} \)
13 \( 1 + 1.21T + 13T^{2} \)
17 \( 1 - 4.08T + 17T^{2} \)
23 \( 1 - 8.95T + 23T^{2} \)
29 \( 1 - 9.38T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 - 3.57T + 41T^{2} \)
43 \( 1 - 7.72T + 43T^{2} \)
47 \( 1 + 9.46T + 47T^{2} \)
53 \( 1 - 11.9T + 53T^{2} \)
59 \( 1 + 7.21T + 59T^{2} \)
61 \( 1 + 4.87T + 61T^{2} \)
67 \( 1 - 11.3T + 67T^{2} \)
71 \( 1 - 9.02T + 71T^{2} \)
73 \( 1 - 5.65T + 73T^{2} \)
79 \( 1 + 9.57T + 79T^{2} \)
83 \( 1 - 10.7T + 83T^{2} \)
89 \( 1 - 11.0T + 89T^{2} \)
97 \( 1 + 8.59T + 97T^{2} \)
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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

−9.555474715908060289694622067948, −8.675914945144312149896043017430, −8.111493524236244550195408506123, −7.54126444756986047731478531029, −6.59969434494003073522238608640, −5.28318340870946592713702623511, −4.50859719447397886383825145621, −3.51121577448641004749618456170, −2.69025241550367804267563357782, −0.930081192474243944491813629973, 0.930081192474243944491813629973, 2.69025241550367804267563357782, 3.51121577448641004749618456170, 4.50859719447397886383825145621, 5.28318340870946592713702623511, 6.59969434494003073522238608640, 7.54126444756986047731478531029, 8.111493524236244550195408506123, 8.675914945144312149896043017430, 9.555474715908060289694622067948

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