Properties

Label 2-2646-1.1-c1-0-29
Degree $2$
Conductor $2646$
Sign $1$
Analytic cond. $21.1284$
Root an. cond. $4.59656$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 3·5-s + 8-s + 3·10-s − 4·13-s + 16-s + 6·17-s − 4·19-s + 3·20-s + 6·23-s + 4·25-s − 4·26-s − 3·29-s + 8·31-s + 32-s + 6·34-s + 8·37-s − 4·38-s + 3·40-s − 6·41-s + 8·43-s + 6·46-s + 6·47-s + 4·50-s − 4·52-s + 9·53-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 1.34·5-s + 0.353·8-s + 0.948·10-s − 1.10·13-s + 1/4·16-s + 1.45·17-s − 0.917·19-s + 0.670·20-s + 1.25·23-s + 4/5·25-s − 0.784·26-s − 0.557·29-s + 1.43·31-s + 0.176·32-s + 1.02·34-s + 1.31·37-s − 0.648·38-s + 0.474·40-s − 0.937·41-s + 1.21·43-s + 0.884·46-s + 0.875·47-s + 0.565·50-s − 0.554·52-s + 1.23·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(21.1284\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2646,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.799773530\)
\(L(\frac12)\) \(\approx\) \(3.799773530\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 - 3 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 - 17 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−9.043257191829633967662307886055, −7.936219177745849931006978485567, −7.20156384395674246812312096130, −6.34847494997515706570624709902, −5.70540244651225272419030082839, −5.06588366539842132791697411033, −4.22943681392943532223456737695, −2.94884338342487937842043672252, −2.36011481689147717993766760666, −1.19829053859815870388190777931, 1.19829053859815870388190777931, 2.36011481689147717993766760666, 2.94884338342487937842043672252, 4.22943681392943532223456737695, 5.06588366539842132791697411033, 5.70540244651225272419030082839, 6.34847494997515706570624709902, 7.20156384395674246812312096130, 7.936219177745849931006978485567, 9.043257191829633967662307886055

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