Properties

Label 2-1716-1.1-c1-0-19
Degree $2$
Conductor $1716$
Sign $-1$
Analytic cond. $13.7023$
Root an. cond. $3.70166$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 0.585·5-s − 2.82·7-s + 9-s − 11-s − 13-s − 0.585·15-s + 2.24·17-s − 2.82·21-s + 0.828·23-s − 4.65·25-s + 27-s − 3.41·29-s − 2.58·31-s − 33-s + 1.65·35-s − 4.82·37-s − 39-s − 7.65·41-s + 8.24·43-s − 0.585·45-s − 8·47-s + 1.00·49-s + 2.24·51-s − 13.3·53-s + 0.585·55-s − 2.82·59-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.261·5-s − 1.06·7-s + 0.333·9-s − 0.301·11-s − 0.277·13-s − 0.151·15-s + 0.543·17-s − 0.617·21-s + 0.172·23-s − 0.931·25-s + 0.192·27-s − 0.634·29-s − 0.464·31-s − 0.174·33-s + 0.280·35-s − 0.793·37-s − 0.160·39-s − 1.19·41-s + 1.25·43-s − 0.0873·45-s − 1.16·47-s + 0.142·49-s + 0.314·51-s − 1.82·53-s + 0.0789·55-s − 0.368·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1716 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1716 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1716\)    =    \(2^{2} \cdot 3 \cdot 11 \cdot 13\)
Sign: $-1$
Analytic conductor: \(13.7023\)
Root analytic conductor: \(3.70166\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1716} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1716,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 - T \)
11 \( 1 + T \)
13 \( 1 + T \)
good5 \( 1 + 0.585T + 5T^{2} \)
7 \( 1 + 2.82T + 7T^{2} \)
17 \( 1 - 2.24T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 0.828T + 23T^{2} \)
29 \( 1 + 3.41T + 29T^{2} \)
31 \( 1 + 2.58T + 31T^{2} \)
37 \( 1 + 4.82T + 37T^{2} \)
41 \( 1 + 7.65T + 41T^{2} \)
43 \( 1 - 8.24T + 43T^{2} \)
47 \( 1 + 8T + 47T^{2} \)
53 \( 1 + 13.3T + 53T^{2} \)
59 \( 1 + 2.82T + 59T^{2} \)
61 \( 1 + 8.82T + 61T^{2} \)
67 \( 1 + 4.24T + 67T^{2} \)
71 \( 1 + 5.17T + 71T^{2} \)
73 \( 1 - 5.65T + 73T^{2} \)
79 \( 1 - 7.07T + 79T^{2} \)
83 \( 1 + 6.34T + 83T^{2} \)
89 \( 1 - 7.89T + 89T^{2} \)
97 \( 1 + 7.17T + 97T^{2} \)
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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

−9.072973926620682511713090167805, −8.049641145042043086743187620029, −7.47234145826826407411344162030, −6.60577758509284272645584940492, −5.75101791819044595225527520120, −4.71468428208679951375031585246, −3.59522428622544930166536907514, −3.05773656710589124985044704637, −1.80198998077609674783874483090, 0, 1.80198998077609674783874483090, 3.05773656710589124985044704637, 3.59522428622544930166536907514, 4.71468428208679951375031585246, 5.75101791819044595225527520120, 6.60577758509284272645584940492, 7.47234145826826407411344162030, 8.049641145042043086743187620029, 9.072973926620682511713090167805

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