Properties

Label 2-1216-1.1-c1-0-4
Degree $2$
Conductor $1216$
Sign $1$
Analytic cond. $9.70980$
Root an. cond. $3.11605$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 5-s − 3·7-s + 9-s − 5·11-s + 4·13-s − 2·15-s − 3·17-s + 19-s + 6·21-s + 8·23-s − 4·25-s + 4·27-s + 2·29-s + 4·31-s + 10·33-s − 3·35-s − 10·37-s − 8·39-s + 10·41-s − 43-s + 45-s − 47-s + 2·49-s + 6·51-s + 4·53-s − 5·55-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.447·5-s − 1.13·7-s + 1/3·9-s − 1.50·11-s + 1.10·13-s − 0.516·15-s − 0.727·17-s + 0.229·19-s + 1.30·21-s + 1.66·23-s − 4/5·25-s + 0.769·27-s + 0.371·29-s + 0.718·31-s + 1.74·33-s − 0.507·35-s − 1.64·37-s − 1.28·39-s + 1.56·41-s − 0.152·43-s + 0.149·45-s − 0.145·47-s + 2/7·49-s + 0.840·51-s + 0.549·53-s − 0.674·55-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: yes
Arithmetic: yes
Character: $\chi_{1216} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7851853327\)
\(L(\frac12)\) \(\approx\) \(0.7851853327\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
5 \( 1 - T + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 13 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 - 9 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−9.888879664431763319657691324377, −9.038018434905016884851105347469, −8.121865683855346250917871835791, −6.89999375237673531400272497166, −6.35171638937504994192666881666, −5.55162300458149510222680140561, −4.93085486739421392203871722376, −3.51647821375355615052320307776, −2.48496537170117614857619865014, −0.67525660115678139253233977173, 0.67525660115678139253233977173, 2.48496537170117614857619865014, 3.51647821375355615052320307776, 4.93085486739421392203871722376, 5.55162300458149510222680140561, 6.35171638937504994192666881666, 6.89999375237673531400272497166, 8.121865683855346250917871835791, 9.038018434905016884851105347469, 9.888879664431763319657691324377

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