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Properties

Label 2-675-27.7-c1-0-17
Degree $2$
Conductor $675$
Sign $0.951 - 0.309i$
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  + (1.57 − 0.574i)2-s + (−1.45 − 0.940i)3-s + (0.632 − 0.530i)4-s + (−2.83 − 0.649i)6-s + (2.99 + 2.51i)7-s + (−0.987 + 1.70i)8-s + (1.23 + 2.73i)9-s + (−0.324 + 1.84i)11-s + (−1.41 + 0.177i)12-s + (−0.688 − 0.250i)13-s + (6.17 + 2.24i)14-s + (−0.862 + 4.89i)16-s + (0.944 + 1.63i)17-s + (3.51 + 3.61i)18-s + (−1.37 + 2.37i)19-s + ⋯
L(s)  = 1  + (1.11 − 0.406i)2-s + (−0.839 − 0.542i)3-s + (0.316 − 0.265i)4-s + (−1.15 − 0.265i)6-s + (1.13 + 0.949i)7-s + (−0.349 + 0.604i)8-s + (0.410 + 0.911i)9-s + (−0.0979 + 0.555i)11-s + (−0.409 + 0.0511i)12-s + (−0.190 − 0.0694i)13-s + (1.65 + 0.600i)14-s + (−0.215 + 1.22i)16-s + (0.229 + 0.396i)17-s + (0.828 + 0.851i)18-s + (−0.314 + 0.544i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 - 0.309i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.951 - 0.309i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $0.951 - 0.309i$
Analytic conductor: \(5.38990\)
Root analytic conductor: \(2.32161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1/2),\ 0.951 - 0.309i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.01071 + 0.318514i\)
\(L(\frac12)\) \(\approx\) \(2.01071 + 0.318514i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.45 + 0.940i)T \)
5 \( 1 \)
good2 \( 1 + (-1.57 + 0.574i)T + (1.53 - 1.28i)T^{2} \)
7 \( 1 + (-2.99 - 2.51i)T + (1.21 + 6.89i)T^{2} \)
11 \( 1 + (0.324 - 1.84i)T + (-10.3 - 3.76i)T^{2} \)
13 \( 1 + (0.688 + 0.250i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (-0.944 - 1.63i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.37 - 2.37i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.46 + 3.74i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (-4.99 + 1.81i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-1.02 + 0.861i)T + (5.38 - 30.5i)T^{2} \)
37 \( 1 + (-1.69 - 2.94i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.68 + 0.614i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (0.873 - 4.95i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (1.30 + 1.09i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + 2.84T + 53T^{2} \)
59 \( 1 + (1.95 + 11.0i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-4.00 - 3.36i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (1.77 + 0.646i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (-6.09 - 10.5i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-4.94 + 8.56i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (11.6 - 4.22i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (10.9 - 3.99i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (-2.86 + 4.96i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.0596 + 0.338i)T + (-91.1 - 33.1i)T^{2} \)
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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

−11.04641762326849176875361370811, −9.992371082134107496271959913766, −8.528286649712114511572717346847, −7.997111298210126572690286911763, −6.66934998753517934447906719204, −5.75203591898589402170430175608, −4.98702185120398133570153159917, −4.40258311566176508114766545210, −2.71593950535104376279588892275, −1.71952124672751709288333241452, 0.923184017089941387691659875856, 3.27317301593646150810046629558, 4.33337933081406546707866225147, 4.91319402488817614164236596568, 5.64042498664549574466411549308, 6.72509537855826955938683468862, 7.42047292592497802369040715589, 8.756803023920985998671559027218, 9.789740481389830156255970221509, 10.69477014162812406236301483256

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