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Properties

Label 2-675-135.34-c1-0-17
Degree $2$
Conductor $675$
Sign $-0.626 - 0.779i$
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  + (0.274 + 0.753i)2-s + (0.386 + 1.68i)3-s + (1.03 − 0.872i)4-s + (−1.16 + 0.753i)6-s + (−1.52 + 1.82i)7-s + (2.33 + 1.34i)8-s + (−2.70 + 1.30i)9-s + (−0.0434 + 0.246i)11-s + (1.87 + 1.41i)12-s + (−0.893 + 2.45i)13-s + (−1.79 − 0.651i)14-s + (0.0969 − 0.549i)16-s + (−0.254 + 0.146i)17-s + (−1.72 − 1.67i)18-s + (−1.39 + 2.41i)19-s + ⋯
L(s)  = 1  + (0.193 + 0.532i)2-s + (0.223 + 0.974i)3-s + (0.519 − 0.436i)4-s + (−0.475 + 0.307i)6-s + (−0.577 + 0.688i)7-s + (0.823 + 0.475i)8-s + (−0.900 + 0.434i)9-s + (−0.0130 + 0.0742i)11-s + (0.541 + 0.409i)12-s + (−0.247 + 0.680i)13-s + (−0.478 − 0.174i)14-s + (0.0242 − 0.137i)16-s + (−0.0616 + 0.0355i)17-s + (−0.406 − 0.395i)18-s + (−0.319 + 0.553i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.626 - 0.779i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.626 - 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $-0.626 - 0.779i$
Analytic conductor: \(5.38990\)
Root analytic conductor: \(2.32161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (574, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1/2),\ -0.626 - 0.779i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.788098 + 1.64362i\)
\(L(\frac12)\) \(\approx\) \(0.788098 + 1.64362i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.386 - 1.68i)T \)
5 \( 1 \)
good2 \( 1 + (-0.274 - 0.753i)T + (-1.53 + 1.28i)T^{2} \)
7 \( 1 + (1.52 - 1.82i)T + (-1.21 - 6.89i)T^{2} \)
11 \( 1 + (0.0434 - 0.246i)T + (-10.3 - 3.76i)T^{2} \)
13 \( 1 + (0.893 - 2.45i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + (0.254 - 0.146i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.39 - 2.41i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.30 - 5.12i)T + (-3.99 + 22.6i)T^{2} \)
29 \( 1 + (0.333 - 0.121i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-2.11 + 1.77i)T + (5.38 - 30.5i)T^{2} \)
37 \( 1 + (6.05 - 3.49i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-9.13 - 3.32i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (0.256 + 0.0452i)T + (40.4 + 14.7i)T^{2} \)
47 \( 1 + (-7.34 + 8.75i)T + (-8.16 - 46.2i)T^{2} \)
53 \( 1 + 5.43iT - 53T^{2} \)
59 \( 1 + (1.03 + 5.88i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (9.07 + 7.61i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (-0.619 + 1.70i)T + (-51.3 - 43.0i)T^{2} \)
71 \( 1 + (-0.185 - 0.320i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-4.35 - 2.51i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.754 + 0.274i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (0.942 + 2.58i)T + (-63.5 + 53.3i)T^{2} \)
89 \( 1 + (-5.22 + 9.05i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (14.6 + 2.57i)T + (91.1 + 33.1i)T^{2} \)
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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

−10.75269268252494792298559095709, −9.826104549113957431035712326401, −9.246053478941880074441600832319, −8.249347543580952462603434917107, −7.17836811000487178903273026357, −6.18876923021997515179360097541, −5.45069207660825748501834939792, −4.54489402113250971781877569798, −3.26851193620954237846935203518, −2.08222401656325163985447692805, 0.883995984913576855660461258456, 2.44429073242986885012074435174, 3.16573281406716012459801119989, 4.35507127683340609149528680613, 5.89844138414546945513703262887, 6.92847764629575195539082057779, 7.33809860965662488323574682132, 8.322412802218244269677455694156, 9.291287633276484961598181379837, 10.63843407794786156237475514121

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