Properties

Label 2-672-1.1-c3-0-33
Degree $2$
Conductor $672$
Sign $-1$
Analytic cond. $39.6492$
Root an. cond. $6.29676$
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  + 3·3-s + 6·5-s − 7·7-s + 9·9-s + 4·11-s − 46·13-s + 18·15-s − 82·17-s − 84·19-s − 21·21-s − 44·23-s − 89·25-s + 27·27-s + 70·29-s + 152·31-s + 12·33-s − 42·35-s − 146·37-s − 138·39-s + 94·41-s − 488·43-s + 54·45-s + 32·47-s + 49·49-s − 246·51-s − 562·53-s + 24·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.536·5-s − 0.377·7-s + 1/3·9-s + 0.109·11-s − 0.981·13-s + 0.309·15-s − 1.16·17-s − 1.01·19-s − 0.218·21-s − 0.398·23-s − 0.711·25-s + 0.192·27-s + 0.448·29-s + 0.880·31-s + 0.0633·33-s − 0.202·35-s − 0.648·37-s − 0.566·39-s + 0.358·41-s − 1.73·43-s + 0.178·45-s + 0.0993·47-s + 1/7·49-s − 0.675·51-s − 1.45·53-s + 0.0588·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $-1$
Analytic conductor: \(39.6492\)
Root analytic conductor: \(6.29676\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 672,\ (\ :3/2),\ -1)\)

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 \( 1 \)
3 \( 1 - p T \)
7 \( 1 + p T \)
good5 \( 1 - 6 T + p^{3} T^{2} \)
11 \( 1 - 4 T + p^{3} T^{2} \)
13 \( 1 + 46 T + p^{3} T^{2} \)
17 \( 1 + 82 T + p^{3} T^{2} \)
19 \( 1 + 84 T + p^{3} T^{2} \)
23 \( 1 + 44 T + p^{3} T^{2} \)
29 \( 1 - 70 T + p^{3} T^{2} \)
31 \( 1 - 152 T + p^{3} T^{2} \)
37 \( 1 + 146 T + p^{3} T^{2} \)
41 \( 1 - 94 T + p^{3} T^{2} \)
43 \( 1 + 488 T + p^{3} T^{2} \)
47 \( 1 - 32 T + p^{3} T^{2} \)
53 \( 1 + 562 T + p^{3} T^{2} \)
59 \( 1 - 476 T + p^{3} T^{2} \)
61 \( 1 - 34 T + p^{3} T^{2} \)
67 \( 1 - 520 T + p^{3} T^{2} \)
71 \( 1 - 36 T + p^{3} T^{2} \)
73 \( 1 + 654 T + p^{3} T^{2} \)
79 \( 1 - 608 T + p^{3} T^{2} \)
83 \( 1 + 284 T + p^{3} T^{2} \)
89 \( 1 + 954 T + p^{3} T^{2} \)
97 \( 1 + 1694 T + p^{3} T^{2} \)
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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

−9.749964559205496294025313011132, −8.827703605635244990058371988036, −8.071769814426987206033840811268, −6.91283022007030543734228684628, −6.27252335664554483381188374415, −4.98132402781375574081806996124, −4.02627801566603567893815822388, −2.71461925792812058284223617379, −1.86518605049888589945042680784, 0, 1.86518605049888589945042680784, 2.71461925792812058284223617379, 4.02627801566603567893815822388, 4.98132402781375574081806996124, 6.27252335664554483381188374415, 6.91283022007030543734228684628, 8.071769814426987206033840811268, 8.827703605635244990058371988036, 9.749964559205496294025313011132

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