Properties

Label 2-1216-1.1-c1-0-35
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.56·3-s − 1.56·5-s − 3·7-s + 3.56·9-s − 3.56·11-s − 2.56·13-s − 4·15-s − 8.12·17-s + 19-s − 7.68·21-s − 1.43·23-s − 2.56·25-s + 1.43·27-s + 7.68·29-s − 0.876·31-s − 9.12·33-s + 4.68·35-s + 1.12·37-s − 6.56·39-s + 4·41-s + 9.56·43-s − 5.56·45-s − 8.68·47-s + 2·49-s − 20.8·51-s + 8.56·53-s + 5.56·55-s + ⋯
L(s)  = 1  + 1.47·3-s − 0.698·5-s − 1.13·7-s + 1.18·9-s − 1.07·11-s − 0.710·13-s − 1.03·15-s − 1.97·17-s + 0.229·19-s − 1.67·21-s − 0.299·23-s − 0.512·25-s + 0.276·27-s + 1.42·29-s − 0.157·31-s − 1.58·33-s + 0.791·35-s + 0.184·37-s − 1.05·39-s + 0.624·41-s + 1.45·43-s − 0.829·45-s − 1.26·47-s + 0.285·49-s − 2.91·51-s + 1.17·53-s + 0.749·55-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: \(1\)
Selberg data: \((2,\ 1216,\ (\ :1/2),\ -1)\)

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 \( 1 \)
19 \( 1 - T \)
good3 \( 1 - 2.56T + 3T^{2} \)
5 \( 1 + 1.56T + 5T^{2} \)
7 \( 1 + 3T + 7T^{2} \)
11 \( 1 + 3.56T + 11T^{2} \)
13 \( 1 + 2.56T + 13T^{2} \)
17 \( 1 + 8.12T + 17T^{2} \)
23 \( 1 + 1.43T + 23T^{2} \)
29 \( 1 - 7.68T + 29T^{2} \)
31 \( 1 + 0.876T + 31T^{2} \)
37 \( 1 - 1.12T + 37T^{2} \)
41 \( 1 - 4T + 41T^{2} \)
43 \( 1 - 9.56T + 43T^{2} \)
47 \( 1 + 8.68T + 47T^{2} \)
53 \( 1 - 8.56T + 53T^{2} \)
59 \( 1 + 8.56T + 59T^{2} \)
61 \( 1 - 5.80T + 61T^{2} \)
67 \( 1 + 4.56T + 67T^{2} \)
71 \( 1 + 12.2T + 71T^{2} \)
73 \( 1 - 7.24T + 73T^{2} \)
79 \( 1 + 10T + 79T^{2} \)
83 \( 1 - 7.36T + 83T^{2} \)
89 \( 1 - 9.36T + 89T^{2} \)
97 \( 1 + 1.12T + 97T^{2} \)
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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

−9.226161577616968284009900057929, −8.526408412427992663998726214108, −7.76902922240119736777179286340, −7.13039175006366889637193122444, −6.15930323606434666381551406606, −4.70974977591438141331643939261, −3.89522227767074913916875693294, −2.87367757091120813503216685532, −2.34497001915234297508936010427, 0, 2.34497001915234297508936010427, 2.87367757091120813503216685532, 3.89522227767074913916875693294, 4.70974977591438141331643939261, 6.15930323606434666381551406606, 7.13039175006366889637193122444, 7.76902922240119736777179286340, 8.526408412427992663998726214108, 9.226161577616968284009900057929

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