Properties

Label 2-1216-1.1-c1-0-23
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  + 2.56·3-s + 4.20·5-s − 1.74·7-s + 3.56·9-s + 6.20·11-s − 4.82·13-s + 10.7·15-s − 5.12·17-s + 19-s − 4.46·21-s − 0.463·23-s + 12.6·25-s + 1.43·27-s − 4.82·29-s + 1.63·31-s + 15.8·33-s − 7.32·35-s − 3.63·37-s − 12.3·39-s − 5.28·41-s + 1.08·43-s + 14.9·45-s + 0.555·47-s − 3.96·49-s − 13.1·51-s − 2.65·53-s + 26.1·55-s + ⋯
L(s)  = 1  + 1.47·3-s + 1.88·5-s − 0.658·7-s + 1.18·9-s + 1.87·11-s − 1.33·13-s + 2.78·15-s − 1.24·17-s + 0.229·19-s − 0.974·21-s − 0.0966·23-s + 2.53·25-s + 0.276·27-s − 0.896·29-s + 0.294·31-s + 2.76·33-s − 1.23·35-s − 0.598·37-s − 1.97·39-s − 0.826·41-s + 0.165·43-s + 2.23·45-s + 0.0809·47-s − 0.566·49-s − 1.83·51-s − 0.365·53-s + 3.51·55-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

\(L(1)\) \(\approx\) \(3.520552292\)
\(L(\frac12)\) \(\approx\) \(3.520552292\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.56T + 3T^{2} \)
5 \( 1 - 4.20T + 5T^{2} \)
7 \( 1 + 1.74T + 7T^{2} \)
11 \( 1 - 6.20T + 11T^{2} \)
13 \( 1 + 4.82T + 13T^{2} \)
17 \( 1 + 5.12T + 17T^{2} \)
23 \( 1 + 0.463T + 23T^{2} \)
29 \( 1 + 4.82T + 29T^{2} \)
31 \( 1 - 1.63T + 31T^{2} \)
37 \( 1 + 3.63T + 37T^{2} \)
41 \( 1 + 5.28T + 41T^{2} \)
43 \( 1 - 1.08T + 43T^{2} \)
47 \( 1 - 0.555T + 47T^{2} \)
53 \( 1 + 2.65T + 53T^{2} \)
59 \( 1 + 1.85T + 59T^{2} \)
61 \( 1 - 9.32T + 61T^{2} \)
67 \( 1 - 4.72T + 67T^{2} \)
71 \( 1 - 11.4T + 71T^{2} \)
73 \( 1 + 0.00646T + 73T^{2} \)
79 \( 1 + 6.76T + 79T^{2} \)
83 \( 1 + 11.8T + 83T^{2} \)
89 \( 1 + 7.31T + 89T^{2} \)
97 \( 1 + 10.4T + 97T^{2} \)
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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

−9.683755030684106525310886187170, −9.112874821974212042669934231190, −8.471669322050260378488247578770, −6.99034840590051602903020580813, −6.68016368521213508621002392349, −5.60729607740898027930719918903, −4.41445634409341867414980068449, −3.30278924354539835556687382360, −2.34536741288858229298641641735, −1.68092204479684283311066936891, 1.68092204479684283311066936891, 2.34536741288858229298641641735, 3.30278924354539835556687382360, 4.41445634409341867414980068449, 5.60729607740898027930719918903, 6.68016368521213508621002392349, 6.99034840590051602903020580813, 8.471669322050260378488247578770, 9.112874821974212042669934231190, 9.683755030684106525310886187170

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