Properties

Label 2-1216-1.1-c3-0-95
Degree $2$
Conductor $1216$
Sign $-1$
Analytic cond. $71.7463$
Root an. cond. $8.47032$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.27·3-s − 1.27·5-s + 12.0·7-s − 8.72·9-s − 18.9·11-s + 55.9·13-s − 5.45·15-s − 89.7·17-s + 19·19-s + 51.7·21-s − 135.·23-s − 123.·25-s − 152.·27-s + 102.·29-s − 103.·31-s − 80.9·33-s − 15.4·35-s − 29.6·37-s + 239.·39-s + 234.·41-s + 53.3·43-s + 11.1·45-s − 33.3·47-s − 196.·49-s − 383.·51-s − 93.1·53-s + 24.1·55-s + ⋯
L(s)  = 1  + 0.822·3-s − 0.114·5-s + 0.653·7-s − 0.323·9-s − 0.518·11-s + 1.19·13-s − 0.0938·15-s − 1.28·17-s + 0.229·19-s + 0.537·21-s − 1.22·23-s − 0.986·25-s − 1.08·27-s + 0.656·29-s − 0.600·31-s − 0.426·33-s − 0.0744·35-s − 0.131·37-s + 0.981·39-s + 0.894·41-s + 0.189·43-s + 0.0368·45-s − 0.103·47-s − 0.573·49-s − 1.05·51-s − 0.241·53-s + 0.0591·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $-1$
Analytic conductor: \(71.7463\)
Root analytic conductor: \(8.47032\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1216,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 - 19T \)
good3 \( 1 - 4.27T + 27T^{2} \)
5 \( 1 + 1.27T + 125T^{2} \)
7 \( 1 - 12.0T + 343T^{2} \)
11 \( 1 + 18.9T + 1.33e3T^{2} \)
13 \( 1 - 55.9T + 2.19e3T^{2} \)
17 \( 1 + 89.7T + 4.91e3T^{2} \)
23 \( 1 + 135.T + 1.21e4T^{2} \)
29 \( 1 - 102.T + 2.43e4T^{2} \)
31 \( 1 + 103.T + 2.97e4T^{2} \)
37 \( 1 + 29.6T + 5.06e4T^{2} \)
41 \( 1 - 234.T + 6.89e4T^{2} \)
43 \( 1 - 53.3T + 7.95e4T^{2} \)
47 \( 1 + 33.3T + 1.03e5T^{2} \)
53 \( 1 + 93.1T + 1.48e5T^{2} \)
59 \( 1 + 637.T + 2.05e5T^{2} \)
61 \( 1 - 125.T + 2.26e5T^{2} \)
67 \( 1 + 119.T + 3.00e5T^{2} \)
71 \( 1 - 18.5T + 3.57e5T^{2} \)
73 \( 1 + 394.T + 3.89e5T^{2} \)
79 \( 1 - 303.T + 4.93e5T^{2} \)
83 \( 1 - 394.T + 5.71e5T^{2} \)
89 \( 1 + 1.02e3T + 7.04e5T^{2} \)
97 \( 1 - 1.32e3T + 9.12e5T^{2} \)
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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

−8.814435689233410324700946369230, −8.155292297111099349349103003784, −7.65911429263614201418208552320, −6.39410044877854407213465872292, −5.62382518223780311882062187912, −4.44772216332919381394772986294, −3.63884067955713425321219261512, −2.54397822723764880614341099712, −1.64569505958463074362590899246, 0, 1.64569505958463074362590899246, 2.54397822723764880614341099712, 3.63884067955713425321219261512, 4.44772216332919381394772986294, 5.62382518223780311882062187912, 6.39410044877854407213465872292, 7.65911429263614201418208552320, 8.155292297111099349349103003784, 8.814435689233410324700946369230

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