Properties

Label 2-3450-1.1-c1-0-37
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 + 0.723·7-s − 8-s + 9-s + 5.51·11-s + 12-s + 4.96·13-s − 0.723·14-s + 16-s + 4.23·17-s − 18-s + 4.96·19-s + 0.723·21-s − 5.51·22-s − 23-s − 24-s − 4.96·26-s + 27-s + 0.723·28-s − 29-s + 6·31-s − 32-s + 5.51·33-s − 4.23·34-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.408·6-s + 0.273·7-s − 0.353·8-s + 0.333·9-s + 1.66·11-s + 0.288·12-s + 1.37·13-s − 0.193·14-s + 0.250·16-s + 1.02·17-s − 0.235·18-s + 1.13·19-s + 0.157·21-s − 1.17·22-s − 0.208·23-s − 0.204·24-s − 0.973·26-s + 0.192·27-s + 0.136·28-s − 0.185·29-s + 1.07·31-s − 0.176·32-s + 0.959·33-s − 0.726·34-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

\(L(1)\) \(\approx\) \(2.314603756\)
\(L(\frac12)\) \(\approx\) \(2.314603756\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.723T + 7T^{2} \)
11 \( 1 - 5.51T + 11T^{2} \)
13 \( 1 - 4.96T + 13T^{2} \)
17 \( 1 - 4.23T + 17T^{2} \)
19 \( 1 - 4.96T + 19T^{2} \)
29 \( 1 + T + 29T^{2} \)
31 \( 1 - 6T + 31T^{2} \)
37 \( 1 + 4.23T + 37T^{2} \)
41 \( 1 - 1.03T + 41T^{2} \)
43 \( 1 + 0.961T + 43T^{2} \)
47 \( 1 + 7.96T + 47T^{2} \)
53 \( 1 + 6.47T + 53T^{2} \)
59 \( 1 + 9.02T + 59T^{2} \)
61 \( 1 - 7.44T + 61T^{2} \)
67 \( 1 + 11.9T + 67T^{2} \)
71 \( 1 + 0.514T + 71T^{2} \)
73 \( 1 + 0.447T + 73T^{2} \)
79 \( 1 + 5.51T + 79T^{2} \)
83 \( 1 - 6.17T + 83T^{2} \)
89 \( 1 + 2.23T + 89T^{2} \)
97 \( 1 - 2.55T + 97T^{2} \)
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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

−8.536940331286590779488832456996, −8.074318729330489632561754665292, −7.25149410515230984303213448128, −6.46530410064055042932591003558, −5.84799120812053353190741778061, −4.66168666350592162631283684863, −3.61571944705244525521802152209, −3.15038335409568857738317489071, −1.62275548728107336960943558507, −1.14556799681141627519721567645, 1.14556799681141627519721567645, 1.62275548728107336960943558507, 3.15038335409568857738317489071, 3.61571944705244525521802152209, 4.66168666350592162631283684863, 5.84799120812053353190741778061, 6.46530410064055042932591003558, 7.25149410515230984303213448128, 8.074318729330489632561754665292, 8.536940331286590779488832456996

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