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Properties

Label 2-675-5.4-c1-0-14
Degree $2$
Conductor $675$
Sign $0.894 - 0.447i$
Analytic cond. $5.38990$
Root an. cond. $2.32161$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.30i·2-s + 0.302·4-s − 4.60i·7-s + 3i·8-s + 2.60·11-s − 0.605i·13-s + 6·14-s − 3.30·16-s − 5.60i·17-s + 3.60·19-s + 3.39i·22-s + 3i·23-s + 0.788·26-s − 1.39i·28-s + 8.60·29-s + ⋯
L(s)  = 1  + 0.921i·2-s + 0.151·4-s − 1.74i·7-s + 1.06i·8-s + 0.785·11-s − 0.167i·13-s + 1.60·14-s − 0.825·16-s − 1.35i·17-s + 0.827·19-s + 0.723i·22-s + 0.625i·23-s + 0.154·26-s − 0.263i·28-s + 1.59·29-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(5.38990\)
Root analytic conductor: \(2.32161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.74399 + 0.411700i\)
\(L(\frac12)\) \(\approx\) \(1.74399 + 0.411700i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 - 1.30iT - 2T^{2} \)
7 \( 1 + 4.60iT - 7T^{2} \)
11 \( 1 - 2.60T + 11T^{2} \)
13 \( 1 + 0.605iT - 13T^{2} \)
17 \( 1 + 5.60iT - 17T^{2} \)
19 \( 1 - 3.60T + 19T^{2} \)
23 \( 1 - 3iT - 23T^{2} \)
29 \( 1 - 8.60T + 29T^{2} \)
31 \( 1 - 1.60T + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 + 2.60T + 41T^{2} \)
43 \( 1 + 6.60iT - 43T^{2} \)
47 \( 1 - 5.21iT - 47T^{2} \)
53 \( 1 - 5.60iT - 53T^{2} \)
59 \( 1 + 8.60T + 59T^{2} \)
61 \( 1 - 10.2T + 61T^{2} \)
67 \( 1 - 15.2iT - 67T^{2} \)
71 \( 1 + 14.6T + 71T^{2} \)
73 \( 1 - 5.39iT - 73T^{2} \)
79 \( 1 - 4.39T + 79T^{2} \)
83 \( 1 + 3iT - 83T^{2} \)
89 \( 1 - 7.81T + 89T^{2} \)
97 \( 1 + 8iT - 97T^{2} \)
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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

−10.50559131195408054611301777794, −9.711178213470994201192700870247, −8.629745812810181199476566922991, −7.50892981610219190339479022645, −7.19212559809768446396074427993, −6.37199828887113485615525922695, −5.19581978869779348902116045603, −4.23657869693161925275712114645, −2.98301382029307851972840356771, −1.11252288828998425687521431192, 1.51638303871680891382246090118, 2.55618720341410673430408508231, 3.50792468847909585478223168385, 4.80931131595874126176538908739, 6.10267676044712679108167173277, 6.60018370978693262080363292424, 8.102919656298651782333163119118, 8.887222839865495159293625659241, 9.678430656907939544900955439648, 10.50014887908914652482624104148

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