Properties

Label 2-6900-1.1-c1-0-19
Degree $2$
Conductor $6900$
Sign $1$
Analytic cond. $55.0967$
Root an. cond. $7.42272$
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  − 3-s + 3·7-s + 9-s − 2·11-s + 2·13-s + 7·17-s − 6·19-s − 3·21-s − 23-s − 27-s − 9·29-s + 9·31-s + 2·33-s + 7·37-s − 2·39-s + 5·41-s − 8·47-s + 2·49-s − 7·51-s + 11·53-s + 6·57-s + 9·59-s + 3·63-s + 3·67-s + 69-s + 3·71-s + 6·73-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.13·7-s + 1/3·9-s − 0.603·11-s + 0.554·13-s + 1.69·17-s − 1.37·19-s − 0.654·21-s − 0.208·23-s − 0.192·27-s − 1.67·29-s + 1.61·31-s + 0.348·33-s + 1.15·37-s − 0.320·39-s + 0.780·41-s − 1.16·47-s + 2/7·49-s − 0.980·51-s + 1.51·53-s + 0.794·57-s + 1.17·59-s + 0.377·63-s + 0.366·67-s + 0.120·69-s + 0.356·71-s + 0.702·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6900\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 23\)
Sign: $1$
Analytic conductor: \(55.0967\)
Root analytic conductor: \(7.42272\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6900,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.918586391\)
\(L(\frac12)\) \(\approx\) \(1.918586391\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
5 \( 1 \)
23 \( 1 + T \)
good7 \( 1 - 3 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 - 9 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 11 T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 3 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 5 T + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−8.069669155801640894146017525841, −7.36913090045138333376970374213, −6.44660273835631511784393340976, −5.73674719055418870802783672106, −5.24541142164515214520039056454, −4.41742680466752310626581711630, −3.77422385356999700348037806420, −2.61537486554548359682980367336, −1.69346935016057027327943982766, −0.76122223760427925475297547796, 0.76122223760427925475297547796, 1.69346935016057027327943982766, 2.61537486554548359682980367336, 3.77422385356999700348037806420, 4.41742680466752310626581711630, 5.24541142164515214520039056454, 5.73674719055418870802783672106, 6.44660273835631511784393340976, 7.36913090045138333376970374213, 8.069669155801640894146017525841

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