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Properties

Label 2-1665-185.184-c1-0-28
Degree $2$
Conductor $1665$
Sign $-i$
Analytic cond. $13.2950$
Root an. cond. $3.64624$
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  − 2·4-s + 2.23i·5-s + 6.08·13-s + 4·16-s − 4.47i·20-s − 5.00·25-s − 2.23i·29-s − 6.08·37-s + 6.08·43-s + 13.6i·47-s + 7·49-s − 12.1·52-s + 13.6i·53-s + 11.1i·59-s − 8·64-s + 13.6i·65-s + ⋯
L(s)  = 1  − 4-s + 0.999i·5-s + 1.68·13-s + 16-s − 0.999i·20-s − 1.00·25-s − 0.415i·29-s − 0.999·37-s + 0.927·43-s + 1.98i·47-s + 49-s − 1.68·52-s + 1.86i·53-s + 1.45i·59-s − 64-s + 1.68i·65-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1665\)    =    \(3^{2} \cdot 5 \cdot 37\)
Sign: $-i$
Analytic conductor: \(13.2950\)
Root analytic conductor: \(3.64624\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1665} (739, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1665,\ (\ :1/2),\ -i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.220902426\)
\(L(\frac12)\) \(\approx\) \(1.220902426\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 - 2.23iT \)
37 \( 1 + 6.08T \)
good2 \( 1 + 2T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 6.08T + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 2.23iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 6.08T + 43T^{2} \)
47 \( 1 - 13.6iT - 47T^{2} \)
53 \( 1 - 13.6iT - 53T^{2} \)
59 \( 1 - 11.1iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 - 13.6iT - 83T^{2} \)
89 \( 1 + 15.6iT - 89T^{2} \)
97 \( 1 + 12.1T + 97T^{2} \)
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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

−9.443331896932609699423299025374, −8.844242195990596399083801502917, −8.028429710292416392857658031932, −7.25252506098691912674179842842, −6.15100583524981028313866403045, −5.70032756860516084449170561286, −4.36719872941918890213512579222, −3.72888508041792395627610472443, −2.78077665358741068544873294678, −1.20581303292556169574097445821, 0.57577531172054496059203774311, 1.69405381104493130178475957333, 3.51279046974039379276444683096, 4.07122233826287886162456495449, 5.11558819864119024947461932841, 5.64435638797109099366972376473, 6.68377866680013896551585309556, 7.907001613848298255566231614178, 8.600354539388376420474085114108, 8.917712065410965219432266107924

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