Properties

Label 2-3450-1.1-c1-0-51
Degree $2$
Conductor $3450$
Sign $-1$
Analytic cond. $27.5483$
Root an. cond. $5.24865$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 6-s + 4.47·7-s − 8-s + 9-s − 5.23·11-s − 12-s − 4.47·13-s − 4.47·14-s + 16-s + 4·17-s − 18-s + 5.70·19-s − 4.47·21-s + 5.23·22-s − 23-s + 24-s + 4.47·26-s − 27-s + 4.47·28-s − 4.47·29-s − 2.47·31-s − 32-s + 5.23·33-s − 4·34-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.408·6-s + 1.69·7-s − 0.353·8-s + 0.333·9-s − 1.57·11-s − 0.288·12-s − 1.24·13-s − 1.19·14-s + 0.250·16-s + 0.970·17-s − 0.235·18-s + 1.30·19-s − 0.975·21-s + 1.11·22-s − 0.208·23-s + 0.204·24-s + 0.877·26-s − 0.192·27-s + 0.845·28-s − 0.830·29-s − 0.444·31-s − 0.176·32-s + 0.911·33-s − 0.685·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3450\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 23\)
Sign: $-1$
Analytic conductor: \(27.5483\)
Root analytic conductor: \(5.24865\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3450,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
5 \( 1 \)
23 \( 1 + T \)
good7 \( 1 - 4.47T + 7T^{2} \)
11 \( 1 + 5.23T + 11T^{2} \)
13 \( 1 + 4.47T + 13T^{2} \)
17 \( 1 - 4T + 17T^{2} \)
19 \( 1 - 5.70T + 19T^{2} \)
29 \( 1 + 4.47T + 29T^{2} \)
31 \( 1 + 2.47T + 31T^{2} \)
37 \( 1 + 11.2T + 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 - 4.76T + 43T^{2} \)
47 \( 1 + 4T + 47T^{2} \)
53 \( 1 + 5.23T + 53T^{2} \)
59 \( 1 + 8.94T + 59T^{2} \)
61 \( 1 - 0.763T + 61T^{2} \)
67 \( 1 + 9.70T + 67T^{2} \)
71 \( 1 - 8.94T + 71T^{2} \)
73 \( 1 - 4.47T + 73T^{2} \)
79 \( 1 - 4.47T + 79T^{2} \)
83 \( 1 + 13.2T + 83T^{2} \)
89 \( 1 + 10.4T + 89T^{2} \)
97 \( 1 + 0.472T + 97T^{2} \)
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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

−7.940872700534058904550003472975, −7.63905764608586473399946470462, −7.15221369035361607506146900366, −5.71362967403371723583071790923, −5.22392005810875671879540815122, −4.75915400332134220466615277380, −3.31280507006915568183011268632, −2.22742130129801859869135830667, −1.37862073498099448356577790316, 0, 1.37862073498099448356577790316, 2.22742130129801859869135830667, 3.31280507006915568183011268632, 4.75915400332134220466615277380, 5.22392005810875671879540815122, 5.71362967403371723583071790923, 7.15221369035361607506146900366, 7.63905764608586473399946470462, 7.940872700534058904550003472975

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