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Properties

Label 2-675-5.4-c3-0-33
Degree $2$
Conductor $675$
Sign $0.447 - 0.894i$
Analytic cond. $39.8262$
Root an. cond. $6.31080$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·4-s + 37i·7-s − 70i·13-s + 64·16-s + 163·19-s + 296i·28-s − 19·31-s + 433i·37-s + 449i·43-s − 1.02e3·49-s − 560i·52-s + 719·61-s + 512·64-s + 880i·67-s − 271i·73-s + ⋯
L(s)  = 1  + 4-s + 1.99i·7-s − 1.49i·13-s + 16-s + 1.96·19-s + 1.99i·28-s − 0.110·31-s + 1.92i·37-s + 1.59i·43-s − 2.99·49-s − 1.49i·52-s + 1.50·61-s + 64-s + 1.60i·67-s − 0.434i·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(39.8262\)
Root analytic conductor: \(6.31080\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :3/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.701008151\)
\(L(\frac12)\) \(\approx\) \(2.701008151\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 - 8T^{2} \)
7 \( 1 - 37iT - 343T^{2} \)
11 \( 1 + 1.33e3T^{2} \)
13 \( 1 + 70iT - 2.19e3T^{2} \)
17 \( 1 - 4.91e3T^{2} \)
19 \( 1 - 163T + 6.85e3T^{2} \)
23 \( 1 - 1.21e4T^{2} \)
29 \( 1 + 2.43e4T^{2} \)
31 \( 1 + 19T + 2.97e4T^{2} \)
37 \( 1 - 433iT - 5.06e4T^{2} \)
41 \( 1 + 6.89e4T^{2} \)
43 \( 1 - 449iT - 7.95e4T^{2} \)
47 \( 1 - 1.03e5T^{2} \)
53 \( 1 - 1.48e5T^{2} \)
59 \( 1 + 2.05e5T^{2} \)
61 \( 1 - 719T + 2.26e5T^{2} \)
67 \( 1 - 880iT - 3.00e5T^{2} \)
71 \( 1 + 3.57e5T^{2} \)
73 \( 1 + 271iT - 3.89e5T^{2} \)
79 \( 1 + 503T + 4.93e5T^{2} \)
83 \( 1 - 5.71e5T^{2} \)
89 \( 1 + 7.04e5T^{2} \)
97 \( 1 - 523iT - 9.12e5T^{2} \)
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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

−10.17086911743834983054082781157, −9.494452201942319938462188342780, −8.355118521535770959554939520664, −7.77881913353151303155891928591, −6.58555901472753891567990605024, −5.65050733363317889129087755091, −5.22771646484566273395918750131, −3.11546848500810103180832529627, −2.71628928804755121360437384943, −1.31892464296328645811167867345, 0.800101163494392080922744910495, 1.87554446615192534822022068435, 3.41007351263224283719934570235, 4.16731921881620373603199531963, 5.46370834143739752590778046173, 6.79041699374872039093559715349, 7.15096164573269241705698939406, 7.84586098052872572990272910799, 9.289817213313369317696058893857, 10.07632922561253161996693268573

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