Properties

Label 2-3450-1.1-c1-0-13
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 6-s + 4.03·7-s − 8-s + 9-s − 5.69·11-s + 12-s + 0.381·13-s − 4.03·14-s + 16-s − 3.65·17-s − 18-s + 0.381·19-s + 4.03·21-s + 5.69·22-s − 23-s − 24-s − 0.381·26-s + 27-s + 4.03·28-s − 29-s + 6·31-s − 32-s − 5.69·33-s + 3.65·34-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.408·6-s + 1.52·7-s − 0.353·8-s + 0.333·9-s − 1.71·11-s + 0.288·12-s + 0.105·13-s − 1.07·14-s + 0.250·16-s − 0.887·17-s − 0.235·18-s + 0.0874·19-s + 0.881·21-s + 1.21·22-s − 0.208·23-s − 0.204·24-s − 0.0747·26-s + 0.192·27-s + 0.763·28-s − 0.185·29-s + 1.07·31-s − 0.176·32-s − 0.991·33-s + 0.627·34-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: $\chi_{3450} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3450,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.808553190\)
\(L(\frac12)\) \(\approx\) \(1.808553190\)
\(L(\frac{3}{2})\) 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 \( 1 - 4.03T + 7T^{2} \)
11 \( 1 + 5.69T + 11T^{2} \)
13 \( 1 - 0.381T + 13T^{2} \)
17 \( 1 + 3.65T + 17T^{2} \)
19 \( 1 - 0.381T + 19T^{2} \)
29 \( 1 + T + 29T^{2} \)
31 \( 1 - 6T + 31T^{2} \)
37 \( 1 - 3.65T + 37T^{2} \)
41 \( 1 - 5.61T + 41T^{2} \)
43 \( 1 - 3.61T + 43T^{2} \)
47 \( 1 + 3.38T + 47T^{2} \)
53 \( 1 - 9.31T + 53T^{2} \)
59 \( 1 - 13.3T + 59T^{2} \)
61 \( 1 - 14.0T + 61T^{2} \)
67 \( 1 + 2.76T + 67T^{2} \)
71 \( 1 - 10.6T + 71T^{2} \)
73 \( 1 + 7.07T + 73T^{2} \)
79 \( 1 - 5.69T + 79T^{2} \)
83 \( 1 - 16.1T + 83T^{2} \)
89 \( 1 - 5.65T + 89T^{2} \)
97 \( 1 + 4.07T + 97T^{2} \)
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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

−8.458236044685757296145522030703, −7.928773252277875989071671914017, −7.54486865720221179721948283738, −6.56512957054520039070410962806, −5.46072757210948399539998628328, −4.84595797655277188728798404537, −3.91798978228615118031293132249, −2.52207169741013342029963023002, −2.19666635096669285790311896893, −0.863877773334512276144341079186, 0.863877773334512276144341079186, 2.19666635096669285790311896893, 2.52207169741013342029963023002, 3.91798978228615118031293132249, 4.84595797655277188728798404537, 5.46072757210948399539998628328, 6.56512957054520039070410962806, 7.54486865720221179721948283738, 7.928773252277875989071671914017, 8.458236044685757296145522030703

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