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Properties

Label 2-675-27.16-c1-0-25
Degree $2$
Conductor $675$
Sign $0.195 + 0.980i$
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  + (0.194 + 0.163i)2-s + (−1.72 − 0.154i)3-s + (−0.336 − 1.90i)4-s + (−0.310 − 0.312i)6-s + (−0.449 + 2.55i)7-s + (0.500 − 0.866i)8-s + (2.95 + 0.534i)9-s + (2.07 − 0.753i)11-s + (0.284 + 3.34i)12-s + (1.11 − 0.932i)13-s + (−0.504 + 0.423i)14-s + (−3.39 + 1.23i)16-s + (1.17 + 2.04i)17-s + (0.487 + 0.586i)18-s + (2.22 − 3.84i)19-s + ⋯
L(s)  = 1  + (0.137 + 0.115i)2-s + (−0.995 − 0.0894i)3-s + (−0.168 − 0.952i)4-s + (−0.126 − 0.127i)6-s + (−0.170 + 0.964i)7-s + (0.176 − 0.306i)8-s + (0.983 + 0.178i)9-s + (0.624 − 0.227i)11-s + (0.0821 + 0.964i)12-s + (0.308 − 0.258i)13-s + (−0.134 + 0.113i)14-s + (−0.849 + 0.309i)16-s + (0.285 + 0.494i)17-s + (0.114 + 0.138i)18-s + (0.509 − 0.882i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.195 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.195 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $0.195 + 0.980i$
Analytic conductor: \(5.38990\)
Root analytic conductor: \(2.32161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1/2),\ 0.195 + 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.784053 - 0.643238i\)
\(L(\frac12)\) \(\approx\) \(0.784053 - 0.643238i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.72 + 0.154i)T \)
5 \( 1 \)
good2 \( 1 + (-0.194 - 0.163i)T + (0.347 + 1.96i)T^{2} \)
7 \( 1 + (0.449 - 2.55i)T + (-6.57 - 2.39i)T^{2} \)
11 \( 1 + (-2.07 + 0.753i)T + (8.42 - 7.07i)T^{2} \)
13 \( 1 + (-1.11 + 0.932i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (-1.17 - 2.04i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.22 + 3.84i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.22 + 6.94i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (4.88 + 4.10i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (1.13 + 6.45i)T + (-29.1 + 10.6i)T^{2} \)
37 \( 1 + (-2.27 - 3.93i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.08 + 4.26i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-5.79 + 2.10i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (-1.65 + 9.40i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + 13.3T + 53T^{2} \)
59 \( 1 + (-3.10 - 1.12i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (1.57 - 8.95i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-4.09 + 3.43i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (1.67 + 2.89i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.55 + 4.41i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.14 - 2.63i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (2.65 + 2.22i)T + (14.4 + 81.7i)T^{2} \)
89 \( 1 + (-7.60 + 13.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.94 + 1.43i)T + (74.3 - 62.3i)T^{2} \)
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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

−10.39744918934205483813671800079, −9.530503556900063064645984536869, −8.814803871909850568426197865957, −7.47015764051502212671678865698, −6.24976474902157164988713953335, −5.99329903209469635659191615209, −5.04329720934670292787636013668, −4.03921045513942661902981737085, −2.17601497153984475711328464633, −0.66083246601424595510569048517, 1.37589929884699535774890583150, 3.44785414113164787540657396006, 4.08812186833448427814192756151, 5.12769052418870091021031391398, 6.26304322549355239681786026948, 7.30846829571090804571771653273, 7.69355779810173534322414381543, 9.208494277818809926620682769788, 9.803406616298174847908309718800, 11.02104816795485390705964086112

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