Properties

Label 2-1216-1.1-c1-0-8
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-s + 4·5-s − 3·7-s − 2·9-s + 2·11-s + 13-s − 4·15-s + 3·17-s − 19-s + 3·21-s + 23-s + 11·25-s + 5·27-s + 5·29-s + 8·31-s − 2·33-s − 12·35-s + 2·37-s − 39-s − 8·41-s + 4·43-s − 8·45-s − 8·47-s + 2·49-s − 3·51-s + 53-s + 8·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.78·5-s − 1.13·7-s − 2/3·9-s + 0.603·11-s + 0.277·13-s − 1.03·15-s + 0.727·17-s − 0.229·19-s + 0.654·21-s + 0.208·23-s + 11/5·25-s + 0.962·27-s + 0.928·29-s + 1.43·31-s − 0.348·33-s − 2.02·35-s + 0.328·37-s − 0.160·39-s − 1.24·41-s + 0.609·43-s − 1.19·45-s − 1.16·47-s + 2/7·49-s − 0.420·51-s + 0.137·53-s + 1.07·55-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

\(L(1)\) \(\approx\) \(1.665917004\)
\(L(\frac12)\) \(\approx\) \(1.665917004\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 - 15 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 3 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 - 9 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−9.958642910212366556628274508824, −9.089395378892366384411617506988, −8.336890921646432967363618675577, −6.67486851525401293685577568233, −6.41602681477419049736409749590, −5.70311644291868843897751049539, −4.89838836690724431377597203004, −3.35219753314436267860210658260, −2.45747388367035991238545338827, −1.03106032747078173831338248002, 1.03106032747078173831338248002, 2.45747388367035991238545338827, 3.35219753314436267860210658260, 4.89838836690724431377597203004, 5.70311644291868843897751049539, 6.41602681477419049736409749590, 6.67486851525401293685577568233, 8.336890921646432967363618675577, 9.089395378892366384411617506988, 9.958642910212366556628274508824

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