Properties

Label 2-3450-1.1-c1-0-25
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 + 4.34·7-s + 8-s + 9-s − 1.07·11-s − 12-s − 2.34·13-s + 4.34·14-s + 16-s + 0.921·17-s + 18-s + 2.34·19-s − 4.34·21-s − 1.07·22-s + 23-s − 24-s − 2.34·26-s − 27-s + 4.34·28-s + 10.4·29-s − 4·31-s + 32-s + 1.07·33-s + 0.921·34-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.408·6-s + 1.64·7-s + 0.353·8-s + 0.333·9-s − 0.325·11-s − 0.288·12-s − 0.649·13-s + 1.15·14-s + 0.250·16-s + 0.223·17-s + 0.235·18-s + 0.536·19-s − 0.947·21-s − 0.229·22-s + 0.208·23-s − 0.204·24-s − 0.458·26-s − 0.192·27-s + 0.820·28-s + 1.94·29-s − 0.718·31-s + 0.176·32-s + 0.187·33-s + 0.158·34-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

\(L(1)\) \(\approx\) \(3.066218080\)
\(L(\frac12)\) \(\approx\) \(3.066218080\)
\(L(\frac{3}{2})\) 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 \( 1 - 4.34T + 7T^{2} \)
11 \( 1 + 1.07T + 11T^{2} \)
13 \( 1 + 2.34T + 13T^{2} \)
17 \( 1 - 0.921T + 17T^{2} \)
19 \( 1 - 2.34T + 19T^{2} \)
29 \( 1 - 10.4T + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 - 2.58T + 37T^{2} \)
41 \( 1 - 0.156T + 41T^{2} \)
43 \( 1 - 0.738T + 43T^{2} \)
47 \( 1 - 6.83T + 47T^{2} \)
53 \( 1 + 0.340T + 53T^{2} \)
59 \( 1 + 8.83T + 59T^{2} \)
61 \( 1 + 11.5T + 61T^{2} \)
67 \( 1 - 2.58T + 67T^{2} \)
71 \( 1 - 15.1T + 71T^{2} \)
73 \( 1 + 4.68T + 73T^{2} \)
79 \( 1 + 11.9T + 79T^{2} \)
83 \( 1 - 11.4T + 83T^{2} \)
89 \( 1 + 4.68T + 89T^{2} \)
97 \( 1 - 9.02T + 97T^{2} \)
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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

−8.361631165081621793738345560813, −7.70041061516606118861642504678, −7.14653540184221539362429971485, −6.15835869421272157305638378557, −5.36320290076653303414790512635, −4.82551895769432277962631371305, −4.29330649336499824513522834704, −3.03290182485077204654775561443, −2.04536548292419714428597429239, −1.02176486620142303189577611982, 1.02176486620142303189577611982, 2.04536548292419714428597429239, 3.03290182485077204654775561443, 4.29330649336499824513522834704, 4.82551895769432277962631371305, 5.36320290076653303414790512635, 6.15835869421272157305638378557, 7.14653540184221539362429971485, 7.70041061516606118861642504678, 8.361631165081621793738345560813

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