Properties

Label 2-1216-1.1-c1-0-30
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 − 3·7-s − 2·9-s + 2·11-s − 13-s − 5·17-s + 19-s − 3·21-s + 23-s − 5·25-s − 5·27-s + 3·29-s − 4·31-s + 2·33-s − 2·37-s − 39-s − 8·41-s − 8·43-s + 8·47-s + 2·49-s − 5·51-s − 9·53-s + 57-s + 59-s − 14·61-s + 6·63-s + 13·67-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.13·7-s − 2/3·9-s + 0.603·11-s − 0.277·13-s − 1.21·17-s + 0.229·19-s − 0.654·21-s + 0.208·23-s − 25-s − 0.962·27-s + 0.557·29-s − 0.718·31-s + 0.348·33-s − 0.328·37-s − 0.160·39-s − 1.24·41-s − 1.21·43-s + 1.16·47-s + 2/7·49-s − 0.700·51-s − 1.23·53-s + 0.132·57-s + 0.130·59-s − 1.79·61-s + 0.755·63-s + 1.58·67-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 \( 1 - T + p T^{2} \)
5 \( 1 + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 13 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 - 9 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 10 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−9.253041138673007375271029562828, −8.699258028013535951563233483351, −7.74515606034087698025852781244, −6.73715959969007181198215210739, −6.17043864352631516446211866530, −5.03277316009204668285879882866, −3.81156163044135505245852369486, −3.09698588528238924660584466755, −2.01374174404058579498750722555, 0, 2.01374174404058579498750722555, 3.09698588528238924660584466755, 3.81156163044135505245852369486, 5.03277316009204668285879882866, 6.17043864352631516446211866530, 6.73715959969007181198215210739, 7.74515606034087698025852781244, 8.699258028013535951563233483351, 9.253041138673007375271029562828

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