Properties

Label 2-1216-1.1-c1-0-16
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  − 2·3-s − 3·5-s + 7-s + 9-s + 3·11-s + 4·13-s + 6·15-s − 3·17-s + 19-s − 2·21-s + 4·25-s + 4·27-s − 6·29-s + 4·31-s − 6·33-s − 3·35-s − 2·37-s − 8·39-s − 6·41-s − 43-s − 3·45-s + 3·47-s − 6·49-s + 6·51-s − 12·53-s − 9·55-s − 2·57-s + ⋯
L(s)  = 1  − 1.15·3-s − 1.34·5-s + 0.377·7-s + 1/3·9-s + 0.904·11-s + 1.10·13-s + 1.54·15-s − 0.727·17-s + 0.229·19-s − 0.436·21-s + 4/5·25-s + 0.769·27-s − 1.11·29-s + 0.718·31-s − 1.04·33-s − 0.507·35-s − 0.328·37-s − 1.28·39-s − 0.937·41-s − 0.152·43-s − 0.447·45-s + 0.437·47-s − 6/7·49-s + 0.840·51-s − 1.64·53-s − 1.21·55-s − 0.264·57-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: yes
Arithmetic: yes
Character: $\chi_{1216} (1, \cdot )$
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 \( 1 + 2 T + p T^{2} \)
5 \( 1 + 3 T + p T^{2} \)
7 \( 1 - T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
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

−9.199158339911279140575946730378, −8.462914393396165009068265832341, −7.65894969899749366452649043261, −6.68601933176938591059893570713, −6.07700564140158959836320975863, −4.96903618429312636799150610379, −4.19306407725294924458030580762, −3.33866387034377157662854676826, −1.39430917521719853589747801849, 0, 1.39430917521719853589747801849, 3.33866387034377157662854676826, 4.19306407725294924458030580762, 4.96903618429312636799150610379, 6.07700564140158959836320975863, 6.68601933176938591059893570713, 7.65894969899749366452649043261, 8.462914393396165009068265832341, 9.199158339911279140575946730378

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