Properties

Label 2-1680-1.1-c3-0-65
Degree $2$
Conductor $1680$
Sign $-1$
Analytic cond. $99.1232$
Root an. cond. $9.95606$
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 + 5·5-s − 7·7-s + 9·9-s − 42·11-s + 20·13-s + 15·15-s + 66·17-s − 38·19-s − 21·21-s − 12·23-s + 25·25-s + 27·27-s − 258·29-s − 146·31-s − 126·33-s − 35·35-s + 434·37-s + 60·39-s − 282·41-s − 20·43-s + 45·45-s + 72·47-s + 49·49-s + 198·51-s + 336·53-s − 210·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 1.15·11-s + 0.426·13-s + 0.258·15-s + 0.941·17-s − 0.458·19-s − 0.218·21-s − 0.108·23-s + 1/5·25-s + 0.192·27-s − 1.65·29-s − 0.845·31-s − 0.664·33-s − 0.169·35-s + 1.92·37-s + 0.246·39-s − 1.07·41-s − 0.0709·43-s + 0.149·45-s + 0.223·47-s + 1/7·49-s + 0.543·51-s + 0.870·53-s − 0.514·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-1$
Analytic conductor: \(99.1232\)
Root analytic conductor: \(9.95606\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1680,\ (\ :3/2),\ -1)\)

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 \( 1 - p T \)
5 \( 1 - p T \)
7 \( 1 + p T \)
good11 \( 1 + 42 T + p^{3} T^{2} \)
13 \( 1 - 20 T + p^{3} T^{2} \)
17 \( 1 - 66 T + p^{3} T^{2} \)
19 \( 1 + 2 p T + p^{3} T^{2} \)
23 \( 1 + 12 T + p^{3} T^{2} \)
29 \( 1 + 258 T + p^{3} T^{2} \)
31 \( 1 + 146 T + p^{3} T^{2} \)
37 \( 1 - 434 T + p^{3} T^{2} \)
41 \( 1 + 282 T + p^{3} T^{2} \)
43 \( 1 + 20 T + p^{3} T^{2} \)
47 \( 1 - 72 T + p^{3} T^{2} \)
53 \( 1 - 336 T + p^{3} T^{2} \)
59 \( 1 - 360 T + p^{3} T^{2} \)
61 \( 1 + 682 T + p^{3} T^{2} \)
67 \( 1 + 812 T + p^{3} T^{2} \)
71 \( 1 + 810 T + p^{3} T^{2} \)
73 \( 1 + 124 T + p^{3} T^{2} \)
79 \( 1 + 1136 T + p^{3} T^{2} \)
83 \( 1 + 156 T + p^{3} T^{2} \)
89 \( 1 + 1038 T + p^{3} T^{2} \)
97 \( 1 - 1208 T + p^{3} T^{2} \)
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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

−8.638307577780672359234010356461, −7.77009943952594211233462479874, −7.21027022986347936511281085244, −6.00533045917114145715677395758, −5.48853096486770319047370649548, −4.31631132439366062534610889767, −3.33312142869497315328442703554, −2.51873453369019607688938975072, −1.47229320978480376234335810280, 0, 1.47229320978480376234335810280, 2.51873453369019607688938975072, 3.33312142869497315328442703554, 4.31631132439366062534610889767, 5.48853096486770319047370649548, 6.00533045917114145715677395758, 7.21027022986347936511281085244, 7.77009943952594211233462479874, 8.638307577780672359234010356461

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