Properties

Label 2-3450-1.1-c1-0-5
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 − 3.12·7-s − 8-s + 9-s − 3.12·11-s + 12-s − 2·13-s + 3.12·14-s + 16-s + 1.12·17-s − 18-s + 4·19-s − 3.12·21-s + 3.12·22-s − 23-s − 24-s + 2·26-s + 27-s − 3.12·28-s + 2·29-s − 32-s − 3.12·33-s − 1.12·34-s + 36-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.408·6-s − 1.18·7-s − 0.353·8-s + 0.333·9-s − 0.941·11-s + 0.288·12-s − 0.554·13-s + 0.834·14-s + 0.250·16-s + 0.272·17-s − 0.235·18-s + 0.917·19-s − 0.681·21-s + 0.665·22-s − 0.208·23-s − 0.204·24-s + 0.392·26-s + 0.192·27-s − 0.590·28-s + 0.371·29-s − 0.176·32-s − 0.543·33-s − 0.192·34-s + 0.166·36-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

\(L(1)\) \(\approx\) \(1.138122157\)
\(L(\frac12)\) \(\approx\) \(1.138122157\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.12T + 7T^{2} \)
11 \( 1 + 3.12T + 11T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 - 1.12T + 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 1.12T + 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 + 12.2T + 53T^{2} \)
59 \( 1 - 2.24T + 59T^{2} \)
61 \( 1 - 9.12T + 61T^{2} \)
67 \( 1 - 8T + 67T^{2} \)
71 \( 1 + 10.2T + 71T^{2} \)
73 \( 1 - 4.24T + 73T^{2} \)
79 \( 1 - 3.12T + 79T^{2} \)
83 \( 1 - 13.3T + 83T^{2} \)
89 \( 1 + 5.12T + 89T^{2} \)
97 \( 1 - 16.2T + 97T^{2} \)
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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

−8.646679262043882079660350934941, −7.81088795689132586530768703778, −7.34914820501121458376115353228, −6.54617218469250239751285841744, −5.72116088804945613648435604920, −4.81152972952805158930851294096, −3.55756712798382439185815253945, −2.93856716003623527264808911580, −2.12528285019996733619865613515, −0.65534543893348699897491693365, 0.65534543893348699897491693365, 2.12528285019996733619865613515, 2.93856716003623527264808911580, 3.55756712798382439185815253945, 4.81152972952805158930851294096, 5.72116088804945613648435604920, 6.54617218469250239751285841744, 7.34914820501121458376115353228, 7.81088795689132586530768703778, 8.646679262043882079660350934941

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