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Properties

Label 2-675-27.13-c1-0-41
Degree $2$
Conductor $675$
Sign $0.410 + 0.911i$
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  + (−0.183 − 1.03i)2-s + (1.72 + 0.0916i)3-s + (0.834 − 0.303i)4-s + (−0.221 − 1.81i)6-s + (2.31 + 0.841i)7-s + (−1.52 − 2.63i)8-s + (2.98 + 0.317i)9-s + (−0.960 + 0.806i)11-s + (1.47 − 0.449i)12-s + (0.789 − 4.47i)13-s + (0.450 − 2.55i)14-s + (−1.09 + 0.921i)16-s + (−3.32 + 5.75i)17-s + (−0.216 − 3.15i)18-s + (−0.124 − 0.215i)19-s + ⋯
L(s)  = 1  + (−0.129 − 0.734i)2-s + (0.998 + 0.0529i)3-s + (0.417 − 0.151i)4-s + (−0.0904 − 0.740i)6-s + (0.873 + 0.317i)7-s + (−0.538 − 0.932i)8-s + (0.994 + 0.105i)9-s + (−0.289 + 0.243i)11-s + (0.424 − 0.129i)12-s + (0.219 − 1.24i)13-s + (0.120 − 0.682i)14-s + (−0.274 + 0.230i)16-s + (−0.806 + 1.39i)17-s + (−0.0511 − 0.743i)18-s + (−0.0285 − 0.0495i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.410 + 0.911i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.410 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $0.410 + 0.911i$
Analytic conductor: \(5.38990\)
Root analytic conductor: \(2.32161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (526, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1/2),\ 0.410 + 0.911i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.02334 - 1.30816i\)
\(L(\frac12)\) \(\approx\) \(2.02334 - 1.30816i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.72 - 0.0916i)T \)
5 \( 1 \)
good2 \( 1 + (0.183 + 1.03i)T + (-1.87 + 0.684i)T^{2} \)
7 \( 1 + (-2.31 - 0.841i)T + (5.36 + 4.49i)T^{2} \)
11 \( 1 + (0.960 - 0.806i)T + (1.91 - 10.8i)T^{2} \)
13 \( 1 + (-0.789 + 4.47i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (3.32 - 5.75i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.124 + 0.215i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.791 + 0.287i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (0.0889 + 0.504i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (-0.770 + 0.280i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (-1.30 + 2.25i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.41 - 8.02i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (3.31 - 2.78i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (4.98 + 1.81i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 - 10.4T + 53T^{2} \)
59 \( 1 + (2.30 + 1.93i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (2.70 + 0.986i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (1.75 - 9.93i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (0.0447 - 0.0774i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.66 + 4.60i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.829 - 4.70i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (1.39 + 7.91i)T + (-77.9 + 28.3i)T^{2} \)
89 \( 1 + (3.35 + 5.80i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.20 - 3.52i)T + (16.8 - 95.5i)T^{2} \)
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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

−10.40966288341064130965102886758, −9.652235490844435743403363089776, −8.525057703606594602590170392811, −8.050064711415615709267310831780, −6.95878615078768098304984953201, −5.84187350218336548442248689792, −4.53917023019370453329439244350, −3.39457580823501698101044623220, −2.41216381335765257080204156107, −1.46954973603837149678952235128, 1.80799319561013551448120132249, 2.83705494853687649893947057972, 4.20569315086647201550393709900, 5.19545567553723160941068419876, 6.65535623013922356007411609087, 7.17110834403588846058581262532, 8.008335050435660978479460835794, 8.732751817430082150758855161880, 9.417687397848734353704295455389, 10.70701447189515495779380158016

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