Properties

Label 2-1216-1.1-c1-0-3
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  − 3·5-s − 5·7-s − 3·9-s + 5·11-s + 4·13-s − 3·17-s − 19-s + 4·25-s + 10·31-s + 15·35-s − 8·37-s + 5·43-s + 9·45-s + 5·47-s + 18·49-s + 6·53-s − 15·55-s + 10·59-s + 5·61-s + 15·63-s − 12·65-s + 10·67-s + 10·71-s − 11·73-s − 25·77-s − 10·79-s + 9·81-s + ⋯
L(s)  = 1  − 1.34·5-s − 1.88·7-s − 9-s + 1.50·11-s + 1.10·13-s − 0.727·17-s − 0.229·19-s + 4/5·25-s + 1.79·31-s + 2.53·35-s − 1.31·37-s + 0.762·43-s + 1.34·45-s + 0.729·47-s + 18/7·49-s + 0.824·53-s − 2.02·55-s + 1.30·59-s + 0.640·61-s + 1.88·63-s − 1.48·65-s + 1.22·67-s + 1.18·71-s − 1.28·73-s − 2.84·77-s − 1.12·79-s + 81-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

\(L(1)\) \(\approx\) \(0.8182032662\)
\(L(\frac12)\) \(\approx\) \(0.8182032662\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
5 \( 1 + 3 T + p T^{2} \)
7 \( 1 + 5 T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 - 5 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 12 T + p T^{2} \)
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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

−9.600208704982168625722141946971, −8.752610644095559283972501030777, −8.410134849556211779065327876559, −6.98286792123187236566096654142, −6.56370906167522668644909842214, −5.76283520417985127913399041730, −4.09350585892641493698723914863, −3.71418757784612877583046899350, −2.78320977751625677782435169813, −0.65261125406809044714433414915, 0.65261125406809044714433414915, 2.78320977751625677782435169813, 3.71418757784612877583046899350, 4.09350585892641493698723914863, 5.76283520417985127913399041730, 6.56370906167522668644909842214, 6.98286792123187236566096654142, 8.410134849556211779065327876559, 8.752610644095559283972501030777, 9.600208704982168625722141946971

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