Properties

Label 2-1216-1.1-c1-0-29
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  + 5-s − 7-s − 3·9-s − 3·11-s + 4·13-s − 3·17-s − 19-s − 8·23-s − 4·25-s + 2·31-s − 35-s + 8·37-s − 11·43-s − 3·45-s − 7·47-s − 6·49-s − 2·53-s − 3·55-s − 6·59-s + 61-s + 3·63-s + 4·65-s + 10·67-s + 2·71-s + 5·73-s + 3·77-s − 2·79-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s − 9-s − 0.904·11-s + 1.10·13-s − 0.727·17-s − 0.229·19-s − 1.66·23-s − 4/5·25-s + 0.359·31-s − 0.169·35-s + 1.31·37-s − 1.67·43-s − 0.447·45-s − 1.02·47-s − 6/7·49-s − 0.274·53-s − 0.404·55-s − 0.781·59-s + 0.128·61-s + 0.377·63-s + 0.496·65-s + 1.22·67-s + 0.237·71-s + 0.585·73-s + 0.341·77-s − 0.225·79-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 \( 1 + p T^{2} \)
5 \( 1 - T + p T^{2} \)
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 \( 1 + 8 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 - 5 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 12 T + p T^{2} \)
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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

−9.433610244190558352178236824457, −8.273077175725298489482911162287, −8.069781997002375745352734843376, −6.52327766267142337926450857689, −6.07380267050193119311144855802, −5.20615537675529298908400867079, −4.00622871486103663130279939128, −2.95678080105101651849410270529, −1.93248528179857739558289949978, 0, 1.93248528179857739558289949978, 2.95678080105101651849410270529, 4.00622871486103663130279939128, 5.20615537675529298908400867079, 6.07380267050193119311144855802, 6.52327766267142337926450857689, 8.069781997002375745352734843376, 8.273077175725298489482911162287, 9.433610244190558352178236824457

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