Properties

Label 2-1216-1.1-c1-0-34
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 7-s − 2·9-s − 6·11-s − 5·13-s + 3·17-s + 19-s + 21-s − 3·23-s − 5·25-s − 5·27-s − 9·29-s + 4·31-s − 6·33-s − 2·37-s − 5·39-s + 8·43-s − 6·49-s + 3·51-s + 3·53-s + 57-s + 9·59-s + 10·61-s − 2·63-s + 5·67-s − 3·69-s + 6·71-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s − 2/3·9-s − 1.80·11-s − 1.38·13-s + 0.727·17-s + 0.229·19-s + 0.218·21-s − 0.625·23-s − 25-s − 0.962·27-s − 1.67·29-s + 0.718·31-s − 1.04·33-s − 0.328·37-s − 0.800·39-s + 1.21·43-s − 6/7·49-s + 0.420·51-s + 0.412·53-s + 0.132·57-s + 1.17·59-s + 1.28·61-s − 0.251·63-s + 0.610·67-s − 0.361·69-s + 0.712·71-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1216,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
5 \( 1 + p T^{2} \)
7 \( 1 - T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−9.488910467190776994037417053794, −8.165660732182309128185574185775, −7.944498198632498175704416365123, −7.14449686564876148233582580493, −5.56559135650201084358170066267, −5.32600053322851104543262796765, −3.98931312476793193392332292270, −2.80468657266648836263703727877, −2.16004950277651645615210004057, 0, 2.16004950277651645615210004057, 2.80468657266648836263703727877, 3.98931312476793193392332292270, 5.32600053322851104543262796765, 5.56559135650201084358170066267, 7.14449686564876148233582580493, 7.944498198632498175704416365123, 8.165660732182309128185574185775, 9.488910467190776994037417053794

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