Properties

Label 2-336-1.1-c3-0-5
Degree $2$
Conductor $336$
Sign $1$
Analytic cond. $19.8246$
Root an. cond. $4.45248$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 14·5-s + 7·7-s + 9·9-s − 4·11-s + 54·13-s − 42·15-s − 14·17-s − 92·19-s − 21·21-s + 152·23-s + 71·25-s − 27·27-s − 106·29-s + 144·31-s + 12·33-s + 98·35-s + 158·37-s − 162·39-s − 390·41-s + 508·43-s + 126·45-s + 528·47-s + 49·49-s + 42·51-s + 606·53-s − 56·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.25·5-s + 0.377·7-s + 1/3·9-s − 0.109·11-s + 1.15·13-s − 0.722·15-s − 0.199·17-s − 1.11·19-s − 0.218·21-s + 1.37·23-s + 0.567·25-s − 0.192·27-s − 0.678·29-s + 0.834·31-s + 0.0633·33-s + 0.473·35-s + 0.702·37-s − 0.665·39-s − 1.48·41-s + 1.80·43-s + 0.417·45-s + 1.63·47-s + 1/7·49-s + 0.115·51-s + 1.57·53-s − 0.137·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $1$
Analytic conductor: \(19.8246\)
Root analytic conductor: \(4.45248\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.113749705\)
\(L(\frac12)\) \(\approx\) \(2.113749705\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + p T \)
7 \( 1 - p T \)
good5 \( 1 - 14 T + p^{3} T^{2} \)
11 \( 1 + 4 T + p^{3} T^{2} \)
13 \( 1 - 54 T + p^{3} T^{2} \)
17 \( 1 + 14 T + p^{3} T^{2} \)
19 \( 1 + 92 T + p^{3} T^{2} \)
23 \( 1 - 152 T + p^{3} T^{2} \)
29 \( 1 + 106 T + p^{3} T^{2} \)
31 \( 1 - 144 T + p^{3} T^{2} \)
37 \( 1 - 158 T + p^{3} T^{2} \)
41 \( 1 + 390 T + p^{3} T^{2} \)
43 \( 1 - 508 T + p^{3} T^{2} \)
47 \( 1 - 528 T + p^{3} T^{2} \)
53 \( 1 - 606 T + p^{3} T^{2} \)
59 \( 1 - 364 T + p^{3} T^{2} \)
61 \( 1 - 678 T + p^{3} T^{2} \)
67 \( 1 + 844 T + p^{3} T^{2} \)
71 \( 1 - 8 T + p^{3} T^{2} \)
73 \( 1 + 422 T + p^{3} T^{2} \)
79 \( 1 + 384 T + p^{3} T^{2} \)
83 \( 1 - 548 T + p^{3} T^{2} \)
89 \( 1 - 1194 T + p^{3} T^{2} \)
97 \( 1 + 1502 T + p^{3} T^{2} \)
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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

−10.91034244295928475835276143207, −10.43212043408376003827637800441, −9.274282088136640568981252223799, −8.491250810901165225797197107152, −7.05145532702476381667586986961, −6.10300272441339421304747494026, −5.39808945544984313617954476785, −4.14545007433564799527712388567, −2.38473203943799878251476206186, −1.09898506836191065390739121338, 1.09898506836191065390739121338, 2.38473203943799878251476206186, 4.14545007433564799527712388567, 5.39808945544984313617954476785, 6.10300272441339421304747494026, 7.05145532702476381667586986961, 8.491250810901165225797197107152, 9.274282088136640568981252223799, 10.43212043408376003827637800441, 10.91034244295928475835276143207

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