Properties

Label 2-1216-1.1-c1-0-12
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 − 2.64·5-s + 0.180·7-s + 3.56·9-s − 0.644·11-s + 3.94·13-s − 6.77·15-s + 5.56·17-s + 19-s + 0.463·21-s + 4.46·23-s + 1.99·25-s + 1.43·27-s + 3.94·29-s + 5.48·31-s − 1.65·33-s − 0.478·35-s − 7.48·37-s + 10.1·39-s + 8.41·41-s − 5.76·43-s − 9.41·45-s + 11.2·47-s − 6.96·49-s + 14.2·51-s − 7.58·53-s + 1.70·55-s + ⋯
L(s)  = 1  + 1.47·3-s − 1.18·5-s + 0.0683·7-s + 1.18·9-s − 0.194·11-s + 1.09·13-s − 1.74·15-s + 1.35·17-s + 0.229·19-s + 0.101·21-s + 0.930·23-s + 0.398·25-s + 0.276·27-s + 0.733·29-s + 0.985·31-s − 0.287·33-s − 0.0808·35-s − 1.23·37-s + 1.61·39-s + 1.31·41-s − 0.879·43-s − 1.40·45-s + 1.64·47-s − 0.995·49-s + 1.99·51-s − 1.04·53-s + 0.229·55-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

\(L(1)\) \(\approx\) \(2.434226596\)
\(L(\frac12)\) \(\approx\) \(2.434226596\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.64T + 5T^{2} \)
7 \( 1 - 0.180T + 7T^{2} \)
11 \( 1 + 0.644T + 11T^{2} \)
13 \( 1 - 3.94T + 13T^{2} \)
17 \( 1 - 5.56T + 17T^{2} \)
23 \( 1 - 4.46T + 23T^{2} \)
29 \( 1 - 3.94T + 29T^{2} \)
31 \( 1 - 5.48T + 31T^{2} \)
37 \( 1 + 7.48T + 37T^{2} \)
41 \( 1 - 8.41T + 41T^{2} \)
43 \( 1 + 5.76T + 43T^{2} \)
47 \( 1 - 11.2T + 47T^{2} \)
53 \( 1 + 7.58T + 53T^{2} \)
59 \( 1 - 11.8T + 59T^{2} \)
61 \( 1 - 2.47T + 61T^{2} \)
67 \( 1 + 8.97T + 67T^{2} \)
71 \( 1 - 7.63T + 71T^{2} \)
73 \( 1 - 10.6T + 73T^{2} \)
79 \( 1 + 10.6T + 79T^{2} \)
83 \( 1 - 5.65T + 83T^{2} \)
89 \( 1 + 17.1T + 89T^{2} \)
97 \( 1 - 3.28T + 97T^{2} \)
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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

−9.542707087762264654244879569118, −8.673286864719519094449754319950, −8.143818285277217066755242438123, −7.63051404706707681983645115393, −6.70881497113278522300333750110, −5.37799706097445190396062840311, −4.16394025055850417656785202711, −3.47557769501295184170228693616, −2.77990280888157109607639758188, −1.19533803010128349088953861527, 1.19533803010128349088953861527, 2.77990280888157109607639758188, 3.47557769501295184170228693616, 4.16394025055850417656785202711, 5.37799706097445190396062840311, 6.70881497113278522300333750110, 7.63051404706707681983645115393, 8.143818285277217066755242438123, 8.673286864719519094449754319950, 9.542707087762264654244879569118

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