Properties

Label 2-1216-1.1-c1-0-15
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  − 3·3-s − 7-s + 6·9-s − 2·11-s + 13-s + 3·17-s + 19-s + 3·21-s + 3·23-s − 5·25-s − 9·27-s − 3·29-s + 8·31-s + 6·33-s + 10·37-s − 3·39-s − 12·41-s − 8·43-s − 8·47-s − 6·49-s − 9·51-s + 9·53-s − 3·57-s + 5·59-s − 10·61-s − 6·63-s − 7·67-s + ⋯
L(s)  = 1  − 1.73·3-s − 0.377·7-s + 2·9-s − 0.603·11-s + 0.277·13-s + 0.727·17-s + 0.229·19-s + 0.654·21-s + 0.625·23-s − 25-s − 1.73·27-s − 0.557·29-s + 1.43·31-s + 1.04·33-s + 1.64·37-s − 0.480·39-s − 1.87·41-s − 1.21·43-s − 1.16·47-s − 6/7·49-s − 1.26·51-s + 1.23·53-s − 0.397·57-s + 0.650·59-s − 1.28·61-s − 0.755·63-s − 0.855·67-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 \( 1 + p T + p T^{2} \)
5 \( 1 + p T^{2} \)
7 \( 1 + T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 5 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 - T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−9.880157654161737946048525039669, −8.410316400065169823569354161047, −7.47536575646542499937872009352, −6.61303503651954034437075836396, −5.91824238738469752095279165720, −5.22289133265133441977616849928, −4.39011552338323920878309071486, −3.11827862407422559284554596552, −1.37942708011457790071691730372, 0, 1.37942708011457790071691730372, 3.11827862407422559284554596552, 4.39011552338323920878309071486, 5.22289133265133441977616849928, 5.91824238738469752095279165720, 6.61303503651954034437075836396, 7.47536575646542499937872009352, 8.410316400065169823569354161047, 9.880157654161737946048525039669

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