Properties

Label 2-1216-1.1-c1-0-21
Degree $2$
Conductor $1216$
Sign $1$
Analytic cond. $9.70980$
Root an. cond. $3.11605$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.08·3-s − 0.786·5-s + 4.29·7-s + 6.51·9-s + 1.21·11-s − 5.08·13-s − 2.42·15-s − 2.29·17-s + 19-s + 13.2·21-s + 7.67·23-s − 4.38·25-s + 10.8·27-s + 0.489·29-s + 3.74·33-s − 3.38·35-s + 2·37-s − 15.6·39-s − 4.16·41-s − 12.9·43-s − 5.12·45-s − 5.80·47-s + 11.4·49-s − 7.08·51-s − 1.93·53-s − 0.954·55-s + 3.08·57-s + ⋯
L(s)  = 1  + 1.78·3-s − 0.351·5-s + 1.62·7-s + 2.17·9-s + 0.365·11-s − 1.41·13-s − 0.626·15-s − 0.557·17-s + 0.229·19-s + 2.89·21-s + 1.60·23-s − 0.876·25-s + 2.08·27-s + 0.0909·29-s + 0.651·33-s − 0.571·35-s + 0.328·37-s − 2.51·39-s − 0.650·41-s − 1.97·43-s − 0.763·45-s − 0.847·47-s + 1.63·49-s − 0.991·51-s − 0.266·53-s − 0.128·55-s + 0.408·57-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: no
Arithmetic: yes
Character: $\chi_{1216} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.338800608\)
\(L(\frac12)\) \(\approx\) \(3.338800608\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.08T + 3T^{2} \)
5 \( 1 + 0.786T + 5T^{2} \)
7 \( 1 - 4.29T + 7T^{2} \)
11 \( 1 - 1.21T + 11T^{2} \)
13 \( 1 + 5.08T + 13T^{2} \)
17 \( 1 + 2.29T + 17T^{2} \)
23 \( 1 - 7.67T + 23T^{2} \)
29 \( 1 - 0.489T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + 4.16T + 41T^{2} \)
43 \( 1 + 12.9T + 43T^{2} \)
47 \( 1 + 5.80T + 47T^{2} \)
53 \( 1 + 1.93T + 53T^{2} \)
59 \( 1 - 11.0T + 59T^{2} \)
61 \( 1 - 5.38T + 61T^{2} \)
67 \( 1 + 2.48T + 67T^{2} \)
71 \( 1 + 11.7T + 71T^{2} \)
73 \( 1 + 8.46T + 73T^{2} \)
79 \( 1 - 1.83T + 79T^{2} \)
83 \( 1 - 7.02T + 83T^{2} \)
89 \( 1 - 13.7T + 89T^{2} \)
97 \( 1 + 3.57T + 97T^{2} \)
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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

−9.530222174403217617299470884144, −8.732284547293606085633511129629, −8.191261480585018046320310898374, −7.49892888563301805436477418141, −6.91011933181459577598289719745, −5.05491666030481672902755041173, −4.51785979064268742069346790353, −3.45316413083897154696901349520, −2.41582926068248001397405527341, −1.56230628834506061117365327499, 1.56230628834506061117365327499, 2.41582926068248001397405527341, 3.45316413083897154696901349520, 4.51785979064268742069346790353, 5.05491666030481672902755041173, 6.91011933181459577598289719745, 7.49892888563301805436477418141, 8.191261480585018046320310898374, 8.732284547293606085633511129629, 9.530222174403217617299470884144

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