Properties

Label 2-3450-1.1-c1-0-29
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 − 4·7-s − 8-s + 9-s − 12-s − 6.47·13-s + 4·14-s + 16-s + 6.47·17-s − 18-s + 2·19-s + 4·21-s + 23-s + 24-s + 6.47·26-s − 27-s − 4·28-s + 8.47·29-s − 32-s − 6.47·34-s + 36-s + 8.47·37-s − 2·38-s + 6.47·39-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.408·6-s − 1.51·7-s − 0.353·8-s + 0.333·9-s − 0.288·12-s − 1.79·13-s + 1.06·14-s + 0.250·16-s + 1.56·17-s − 0.235·18-s + 0.458·19-s + 0.872·21-s + 0.208·23-s + 0.204·24-s + 1.26·26-s − 0.192·27-s − 0.755·28-s + 1.57·29-s − 0.176·32-s − 1.10·34-s + 0.166·36-s + 1.39·37-s − 0.324·38-s + 1.03·39-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: \(1\)
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 \( 1 + 4T + 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 6.47T + 13T^{2} \)
17 \( 1 - 6.47T + 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
29 \( 1 - 8.47T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 8.47T + 37T^{2} \)
41 \( 1 + 6.94T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 - 12.9T + 47T^{2} \)
53 \( 1 + 8.94T + 53T^{2} \)
59 \( 1 + 6T + 59T^{2} \)
61 \( 1 - 8.47T + 61T^{2} \)
67 \( 1 + 12.9T + 67T^{2} \)
71 \( 1 + 16.4T + 71T^{2} \)
73 \( 1 - 12.9T + 73T^{2} \)
79 \( 1 - 3.52T + 79T^{2} \)
83 \( 1 + 10.4T + 83T^{2} \)
89 \( 1 + 7.52T + 89T^{2} \)
97 \( 1 + 13.4T + 97T^{2} \)
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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

−8.130309638465332528014322246729, −7.37173515845387402954839085114, −6.88375719992568939833887814443, −6.06376656325120808491841765981, −5.39537623854135865466825159426, −4.42101883404161236949210738025, −3.16999488983165092335961262818, −2.64379062423740290989239934204, −1.08526199425932897027075184461, 0, 1.08526199425932897027075184461, 2.64379062423740290989239934204, 3.16999488983165092335961262818, 4.42101883404161236949210738025, 5.39537623854135865466825159426, 6.06376656325120808491841765981, 6.88375719992568939833887814443, 7.37173515845387402954839085114, 8.130309638465332528014322246729

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