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Properties

Label 2-1665-185.73-c0-0-4
Degree $2$
Conductor $1665$
Sign $0.681 - 0.731i$
Analytic cond. $0.830943$
Root an. cond. $0.911560$
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  + (0.785 + 0.785i)2-s + 0.234i·4-s + (−0.831 + 0.555i)5-s + (0.541 − 0.541i)7-s + (0.601 − 0.601i)8-s + (−1.08 − 0.216i)10-s + 0.850·14-s + 1.17·16-s + (1.17 + 1.17i)17-s + (−0.130 − 0.195i)20-s + (−0.275 + 0.275i)23-s + (0.382 − 0.923i)25-s + (0.126 + 0.126i)28-s + 0.390·29-s + (0.325 + 0.325i)32-s + ⋯
L(s)  = 1  + (0.785 + 0.785i)2-s + 0.234i·4-s + (−0.831 + 0.555i)5-s + (0.541 − 0.541i)7-s + (0.601 − 0.601i)8-s + (−1.08 − 0.216i)10-s + 0.850·14-s + 1.17·16-s + (1.17 + 1.17i)17-s + (−0.130 − 0.195i)20-s + (−0.275 + 0.275i)23-s + (0.382 − 0.923i)25-s + (0.126 + 0.126i)28-s + 0.390·29-s + (0.325 + 0.325i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1665 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.681 - 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1665 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.681 - 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1665\)    =    \(3^{2} \cdot 5 \cdot 37\)
Sign: $0.681 - 0.731i$
Analytic conductor: \(0.830943\)
Root analytic conductor: \(0.911560\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1665} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1665,\ (\ :0),\ 0.681 - 0.731i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.663037542\)
\(L(\frac12)\) \(\approx\) \(1.663037542\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (0.831 - 0.555i)T \)
37 \( 1 + (-0.707 + 0.707i)T \)
good2 \( 1 + (-0.785 - 0.785i)T + iT^{2} \)
7 \( 1 + (-0.541 + 0.541i)T - iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + (-1.17 - 1.17i)T + iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + (0.275 - 0.275i)T - iT^{2} \)
29 \( 1 - 0.390T + T^{2} \)
31 \( 1 - T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + 1.96T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + (1.30 - 1.30i)T - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (1.30 + 1.30i)T + iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + 1.11T + T^{2} \)
97 \( 1 + iT^{2} \)
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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

−9.825921278998850397805282516242, −8.521692095169236232343586119793, −7.62958393511938136901042517036, −7.42582561368229360587044368798, −6.31391349650237514926364966062, −5.73334249364526479603683429917, −4.59684064321889145143727665686, −4.04663490855787091530207810020, −3.14994989443707941702475099120, −1.39179995174669800021871636443, 1.36337486740955026041383577639, 2.68714448969150889852724049192, 3.44580391231882145781215977081, 4.51306741342798443651633843935, 4.96702387584230812479305420985, 5.85978998243760471572498979874, 7.28757009420429972206599181677, 7.921875902588193360036365931852, 8.556267122489901198917283168258, 9.492217386175447014694625663863

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