Properties

Label 2-3450-1.1-c1-0-22
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 − 2·7-s + 8-s + 9-s − 4.24·11-s + 12-s − 0.828·13-s − 2·14-s + 16-s + 6.82·17-s + 18-s + 6.24·19-s − 2·21-s − 4.24·22-s − 23-s + 24-s − 0.828·26-s + 27-s − 2·28-s + 3.65·29-s + 6·31-s + 32-s − 4.24·33-s + 6.82·34-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.408·6-s − 0.755·7-s + 0.353·8-s + 0.333·9-s − 1.27·11-s + 0.288·12-s − 0.229·13-s − 0.534·14-s + 0.250·16-s + 1.65·17-s + 0.235·18-s + 1.43·19-s − 0.436·21-s − 0.904·22-s − 0.208·23-s + 0.204·24-s − 0.162·26-s + 0.192·27-s − 0.377·28-s + 0.679·29-s + 1.07·31-s + 0.176·32-s − 0.738·33-s + 1.17·34-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

\(L(1)\) \(\approx\) \(3.451492090\)
\(L(\frac12)\) \(\approx\) \(3.451492090\)
\(L(\frac{3}{2})\) 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 \( 1 + 2T + 7T^{2} \)
11 \( 1 + 4.24T + 11T^{2} \)
13 \( 1 + 0.828T + 13T^{2} \)
17 \( 1 - 6.82T + 17T^{2} \)
19 \( 1 - 6.24T + 19T^{2} \)
29 \( 1 - 3.65T + 29T^{2} \)
31 \( 1 - 6T + 31T^{2} \)
37 \( 1 - 0.585T + 37T^{2} \)
41 \( 1 + 6.82T + 41T^{2} \)
43 \( 1 - 10.2T + 43T^{2} \)
47 \( 1 + 0.828T + 47T^{2} \)
53 \( 1 - 10.5T + 53T^{2} \)
59 \( 1 - 8.48T + 59T^{2} \)
61 \( 1 + 0.585T + 61T^{2} \)
67 \( 1 + 3.41T + 67T^{2} \)
71 \( 1 + 5.65T + 71T^{2} \)
73 \( 1 - 7.65T + 73T^{2} \)
79 \( 1 + 3.65T + 79T^{2} \)
83 \( 1 - 1.41T + 83T^{2} \)
89 \( 1 - 9.17T + 89T^{2} \)
97 \( 1 + 6T + 97T^{2} \)
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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

−8.413774508242602887310055027479, −7.69253562486559314647371303774, −7.25676741832027207040146859140, −6.23485547883937440421083290152, −5.46422237695515425930360580867, −4.86769104484616890814384966162, −3.70642958831923559239424793583, −3.06609435068249276371982559976, −2.46249404570952105233852741483, −0.984142744538391190326933600485, 0.984142744538391190326933600485, 2.46249404570952105233852741483, 3.06609435068249276371982559976, 3.70642958831923559239424793583, 4.86769104484616890814384966162, 5.46422237695515425930360580867, 6.23485547883937440421083290152, 7.25676741832027207040146859140, 7.69253562486559314647371303774, 8.413774508242602887310055027479

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