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Properties

Label 2-675-27.13-c1-0-49
Degree $2$
Conductor $675$
Sign $-0.0581 - 0.998i$
Analytic cond. $5.38990$
Root an. cond. $2.32161$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.233 − 1.32i)2-s + (−1.70 − 0.300i)3-s + (0.173 − 0.0632i)4-s + 2.33i·6-s + (−0.652 − 0.237i)7-s + (−1.47 − 2.54i)8-s + (2.81 + 1.02i)9-s + (−3.52 + 2.95i)11-s + (−0.315 + 0.0555i)12-s + (0.245 − 1.39i)13-s + (−0.162 + 0.921i)14-s + (−2.75 + 2.31i)16-s + (−1.93 + 3.35i)17-s + (0.701 − 3.98i)18-s + (−3.53 − 6.11i)19-s + ⋯
L(s)  = 1  + (−0.165 − 0.938i)2-s + (−0.984 − 0.173i)3-s + (0.0868 − 0.0316i)4-s + 0.952i·6-s + (−0.246 − 0.0897i)7-s + (−0.520 − 0.901i)8-s + (0.939 + 0.342i)9-s + (−1.06 + 0.890i)11-s + (−0.0909 + 0.0160i)12-s + (0.0679 − 0.385i)13-s + (−0.0434 + 0.246i)14-s + (−0.688 + 0.577i)16-s + (−0.470 + 0.814i)17-s + (0.165 − 0.938i)18-s + (−0.810 − 1.40i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0581 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0581 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $-0.0581 - 0.998i$
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: \(1\)
Selberg data: \((2,\ 675,\ (\ :1/2),\ -0.0581 - 0.998i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.70 + 0.300i)T \)
5 \( 1 \)
good2 \( 1 + (0.233 + 1.32i)T + (-1.87 + 0.684i)T^{2} \)
7 \( 1 + (0.652 + 0.237i)T + (5.36 + 4.49i)T^{2} \)
11 \( 1 + (3.52 - 2.95i)T + (1.91 - 10.8i)T^{2} \)
13 \( 1 + (-0.245 + 1.39i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (1.93 - 3.35i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.53 + 6.11i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.59 - 1.30i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (-0.851 - 4.82i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (-0.786 + 0.286i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (3.99 - 6.91i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.36 - 7.74i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (-1.59 + 1.33i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (-6.46 - 2.35i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + 3.05T + 53T^{2} \)
59 \( 1 + (6.82 + 5.72i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-8.12 - 2.95i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (1.64 - 9.30i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-2.90 + 5.02i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.70 - 4.68i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.27 + 12.9i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-0.197 - 1.11i)T + (-77.9 + 28.3i)T^{2} \)
89 \( 1 + (0.368 + 0.637i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.39 + 5.36i)T + (16.8 - 95.5i)T^{2} \)
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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

−10.32494002486971960104497934544, −9.407835912259115072033063833523, −8.109810862631563741813452070703, −6.96698570650909170308940011581, −6.39227331413677401512499908302, −5.20918963466175347616839827450, −4.22716565122897699184800277911, −2.77004168902742188106429922047, −1.64991004072917484349343518626, 0, 2.32926179299198265274773399911, 3.90998255341714336243277528115, 5.21281623832183930590167215293, 5.92213851398612515515146254610, 6.54527740650518425783630057157, 7.54704229212666484988879616125, 8.303742413722767247512539950792, 9.329894984036971959745290937872, 10.45295102832225454660073648605

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