Properties

Label 2-1216-1.1-c3-0-84
Degree $2$
Conductor $1216$
Sign $-1$
Analytic cond. $71.7463$
Root an. cond. $8.47032$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 8·5-s + 17·7-s − 26·9-s − 70·11-s + 61·13-s − 8·15-s + 83·17-s + 19·19-s − 17·21-s − 115·23-s − 61·25-s + 53·27-s − 279·29-s + 72·31-s + 70·33-s + 136·35-s + 34·37-s − 61·39-s + 108·41-s − 192·43-s − 208·45-s + 392·47-s − 54·49-s − 83·51-s − 131·53-s − 560·55-s + ⋯
L(s)  = 1  − 0.192·3-s + 0.715·5-s + 0.917·7-s − 0.962·9-s − 1.91·11-s + 1.30·13-s − 0.137·15-s + 1.18·17-s + 0.229·19-s − 0.176·21-s − 1.04·23-s − 0.487·25-s + 0.377·27-s − 1.78·29-s + 0.417·31-s + 0.369·33-s + 0.656·35-s + 0.151·37-s − 0.250·39-s + 0.411·41-s − 0.680·43-s − 0.689·45-s + 1.21·47-s − 0.157·49-s − 0.227·51-s − 0.339·53-s − 1.37·55-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: yes
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 \( 1 - p T \)
good3 \( 1 + T + p^{3} T^{2} \)
5 \( 1 - 8 T + p^{3} T^{2} \)
7 \( 1 - 17 T + p^{3} T^{2} \)
11 \( 1 + 70 T + p^{3} T^{2} \)
13 \( 1 - 61 T + p^{3} T^{2} \)
17 \( 1 - 83 T + p^{3} T^{2} \)
23 \( 1 + 5 p T + p^{3} T^{2} \)
29 \( 1 + 279 T + p^{3} T^{2} \)
31 \( 1 - 72 T + p^{3} T^{2} \)
37 \( 1 - 34 T + p^{3} T^{2} \)
41 \( 1 - 108 T + p^{3} T^{2} \)
43 \( 1 + 192 T + p^{3} T^{2} \)
47 \( 1 - 392 T + p^{3} T^{2} \)
53 \( 1 + 131 T + p^{3} T^{2} \)
59 \( 1 + 609 T + p^{3} T^{2} \)
61 \( 1 + 338 T + p^{3} T^{2} \)
67 \( 1 + 461 T + p^{3} T^{2} \)
71 \( 1 + 750 T + p^{3} T^{2} \)
73 \( 1 - 1177 T + p^{3} T^{2} \)
79 \( 1 - 22 T + p^{3} T^{2} \)
83 \( 1 + 810 T + p^{3} T^{2} \)
89 \( 1 + 476 T + p^{3} T^{2} \)
97 \( 1 - 1426 T + p^{3} T^{2} \)
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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

−8.877689412732737794514107895669, −7.948930520487697621605378567853, −7.69684493300998657292965082617, −5.94750298207995588932253930595, −5.74828005423672165153507695180, −4.92169640887164943393324572291, −3.55455983813344743745970485696, −2.49516331906696751009521800059, −1.48142883843209680048832769095, 0, 1.48142883843209680048832769095, 2.49516331906696751009521800059, 3.55455983813344743745970485696, 4.92169640887164943393324572291, 5.74828005423672165153507695180, 5.94750298207995588932253930595, 7.69684493300998657292965082617, 7.948930520487697621605378567853, 8.877689412732737794514107895669

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