Properties

Label 2-1216-1.1-c1-0-13
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  + 2·3-s + 5-s − 3·7-s + 9-s + 3·11-s + 4·13-s + 2·15-s + 5·17-s + 19-s − 6·21-s − 4·25-s − 4·27-s − 2·29-s + 8·31-s + 6·33-s − 3·35-s + 10·37-s + 8·39-s + 6·41-s + 7·43-s + 45-s − 9·47-s + 2·49-s + 10·51-s + 8·53-s + 3·55-s + 2·57-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.447·5-s − 1.13·7-s + 1/3·9-s + 0.904·11-s + 1.10·13-s + 0.516·15-s + 1.21·17-s + 0.229·19-s − 1.30·21-s − 4/5·25-s − 0.769·27-s − 0.371·29-s + 1.43·31-s + 1.04·33-s − 0.507·35-s + 1.64·37-s + 1.28·39-s + 0.937·41-s + 1.06·43-s + 0.149·45-s − 1.31·47-s + 2/7·49-s + 1.40·51-s + 1.09·53-s + 0.404·55-s + 0.264·57-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: 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

\(L(1)\) \(\approx\) \(2.577861644\)
\(L(\frac12)\) \(\approx\) \(2.577861644\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
5 \( 1 - T + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 7 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 - 8 T + p T^{2} \)
59 \( 1 + 14 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 15 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 16 T + p T^{2} \)
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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

−9.563320344627364655050372899522, −9.075343088620985226404492276671, −8.186562113914066026677579677805, −7.43537131861627444263317650332, −6.20879084669067007942487517486, −5.89904763388088518363904025264, −4.19084046620099313866498183608, −3.41145508666670040670114085640, −2.67622771685970545269516699451, −1.26674131115266326801058270565, 1.26674131115266326801058270565, 2.67622771685970545269516699451, 3.41145508666670040670114085640, 4.19084046620099313866498183608, 5.89904763388088518363904025264, 6.20879084669067007942487517486, 7.43537131861627444263317650332, 8.186562113914066026677579677805, 9.075343088620985226404492276671, 9.563320344627364655050372899522

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