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Properties

Label 2-1375-1375.659-c0-0-0
Degree $2$
Conductor $1375$
Sign $-0.988 - 0.150i$
Analytic cond. $0.686214$
Root an. cond. $0.828380$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.813 − 0.516i)3-s + (−0.0627 − 0.998i)4-s + (−0.309 − 0.951i)5-s + (−0.0305 − 0.0648i)9-s + (−0.728 − 0.684i)11-s + (−0.464 + 0.844i)12-s + (−0.239 + 0.933i)15-s + (−0.992 + 0.125i)16-s + (−0.929 + 0.368i)20-s + (1.65 − 0.316i)23-s + (−0.809 + 0.587i)25-s + (−0.129 + 1.02i)27-s + (0.0235 − 0.374i)31-s + (0.239 + 0.933i)33-s + (−0.0627 + 0.0345i)36-s + (0.147 + 1.16i)37-s + ⋯
L(s)  = 1  + (−0.813 − 0.516i)3-s + (−0.0627 − 0.998i)4-s + (−0.309 − 0.951i)5-s + (−0.0305 − 0.0648i)9-s + (−0.728 − 0.684i)11-s + (−0.464 + 0.844i)12-s + (−0.239 + 0.933i)15-s + (−0.992 + 0.125i)16-s + (−0.929 + 0.368i)20-s + (1.65 − 0.316i)23-s + (−0.809 + 0.587i)25-s + (−0.129 + 1.02i)27-s + (0.0235 − 0.374i)31-s + (0.239 + 0.933i)33-s + (−0.0627 + 0.0345i)36-s + (0.147 + 1.16i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $-0.988 - 0.150i$
Analytic conductor: \(0.686214\)
Root analytic conductor: \(0.828380\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (659, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1375,\ (\ :0),\ -0.988 - 0.150i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5283024608\)
\(L(\frac12)\) \(\approx\) \(0.5283024608\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (0.728 + 0.684i)T \)
good2 \( 1 + (0.0627 + 0.998i)T^{2} \)
3 \( 1 + (0.813 + 0.516i)T + (0.425 + 0.904i)T^{2} \)
7 \( 1 + (-0.809 + 0.587i)T^{2} \)
13 \( 1 + (-0.637 + 0.770i)T^{2} \)
17 \( 1 + (-0.992 - 0.125i)T^{2} \)
19 \( 1 + (0.425 - 0.904i)T^{2} \)
23 \( 1 + (-1.65 + 0.316i)T + (0.929 - 0.368i)T^{2} \)
29 \( 1 + (-0.728 + 0.684i)T^{2} \)
31 \( 1 + (-0.0235 + 0.374i)T + (-0.992 - 0.125i)T^{2} \)
37 \( 1 + (-0.147 - 1.16i)T + (-0.968 + 0.248i)T^{2} \)
41 \( 1 + (0.929 + 0.368i)T^{2} \)
43 \( 1 + (0.309 - 0.951i)T^{2} \)
47 \( 1 + (1.46 - 1.21i)T + (0.187 - 0.982i)T^{2} \)
53 \( 1 + (-0.340 + 1.32i)T + (-0.876 - 0.481i)T^{2} \)
59 \( 1 + (1.73 + 0.955i)T + (0.535 + 0.844i)T^{2} \)
61 \( 1 + (0.929 - 0.368i)T^{2} \)
67 \( 1 + (0.183 - 0.462i)T + (-0.728 - 0.684i)T^{2} \)
71 \( 1 + (1.11 + 1.35i)T + (-0.187 + 0.982i)T^{2} \)
73 \( 1 + (0.535 - 0.844i)T^{2} \)
79 \( 1 + (0.425 + 0.904i)T^{2} \)
83 \( 1 + (-0.425 + 0.904i)T^{2} \)
89 \( 1 + (-1.11 + 0.614i)T + (0.535 - 0.844i)T^{2} \)
97 \( 1 + (-0.0922 - 0.233i)T + (-0.728 + 0.684i)T^{2} \)
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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

−9.294412650917021845375416726067, −8.671377737721794328164331046536, −7.71482829570952805746224556146, −6.66737998491479990480300234861, −5.99255433194192141845749846616, −5.19214308154849303764603221903, −4.68515772643693772127661498050, −3.15482404923116588115953900085, −1.51951211433278030176650159843, −0.49724346505651285665227898390, 2.41592675137321231807166500876, 3.25782998453288174721859298635, 4.32179112484362917814291790296, 5.05772398348961416430540371854, 6.09290936092251671147543907309, 7.19948514619512099066977443131, 7.50764595646845755672985780687, 8.540337661956293614432373583958, 9.482599348338123289951063939317, 10.50050411417949450861811826437

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