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Properties

Label 2-675-15.14-c0-0-0
Degree $2$
Conductor $675$
Sign $0.447 - 0.894i$
Analytic cond. $0.336868$
Root an. cond. $0.580404$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s + i·7-s + 2i·13-s + 16-s + 19-s − i·28-s − 31-s + i·37-s − i·43-s − 2i·52-s − 61-s − 64-s − 2i·67-s − i·73-s − 76-s + ⋯
L(s)  = 1  − 4-s + i·7-s + 2i·13-s + 16-s + 19-s − i·28-s − 31-s + i·37-s − i·43-s − 2i·52-s − 61-s − 64-s − 2i·67-s − i·73-s − 76-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(0.336868\)
Root analytic conductor: \(0.580404\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (674, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :0),\ 0.447 - 0.894i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7248099457\)
\(L(\frac12)\) \(\approx\) \(0.7248099457\)
\(L(1)\) 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 \( 1 + T^{2} \)
7 \( 1 - iT - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - 2iT - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T + T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + T + T^{2} \)
37 \( 1 - iT - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + iT - T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T + T^{2} \)
67 \( 1 + 2iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + iT - T^{2} \)
79 \( 1 - T + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - iT - T^{2} \)
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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

−10.88001516629769638639021366239, −9.615608100842875545926687207280, −9.217022821958668079393325244446, −8.531185975237468514862768687538, −7.42626025789146920812590421178, −6.32122651159402923306347575626, −5.32222385461243599161022573612, −4.49721641591086906274051710305, −3.39282298873846328586922159066, −1.84901785243619914988913026027, 0.903546389881432883975382087892, 3.11645252466251850056974162544, 3.98437199804989189250443829933, 5.11346732251850265724749373303, 5.82383502029821587873126107837, 7.35532613837584218097585801942, 7.86139484924557572318788804059, 8.845288129942645932477566539947, 9.831807487832954854800311824728, 10.37597687462054513137667883576

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