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Properties

Label 2-405-45.29-c0-0-1
Degree $2$
Conductor $405$
Sign $0.342 + 0.939i$
Analytic cond. $0.202121$
Root an. cond. $0.449579$
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  + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)5-s + 8-s − 0.999·10-s + (0.5 − 0.866i)16-s − 17-s − 19-s + (0.5 + 0.866i)23-s + (−0.499 + 0.866i)25-s + (0.5 + 0.866i)31-s + (−0.5 + 0.866i)34-s + (−0.5 + 0.866i)38-s + (−0.500 − 0.866i)40-s + 0.999·46-s + (−1 + 1.73i)47-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)5-s + 8-s − 0.999·10-s + (0.5 − 0.866i)16-s − 17-s − 19-s + (0.5 + 0.866i)23-s + (−0.499 + 0.866i)25-s + (0.5 + 0.866i)31-s + (−0.5 + 0.866i)34-s + (−0.5 + 0.866i)38-s + (−0.500 − 0.866i)40-s + 0.999·46-s + (−1 + 1.73i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.342 + 0.939i$
Analytic conductor: \(0.202121\)
Root analytic conductor: \(0.449579\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :0),\ 0.342 + 0.939i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.055880458\)
\(L(\frac12)\) \(\approx\) \(1.055880458\)
\(L(1)\) 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 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 + T + T^{2} \)
19 \( 1 + T + T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + T + T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.5 + 0.866i)T^{2} \)
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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

−11.37914445742390888364191252848, −10.78043409833624124403047556995, −9.569657504300470819616103727247, −8.567094889802317980942082513128, −7.73355615850760394401301738471, −6.57761387172952325273896913430, −5.01996500504280180830195282942, −4.30746236260083919008629593266, −3.20070404887720902396814416450, −1.72241182277389574123742867895, 2.36470940730744213209832599340, 3.96283284314292897956239151632, 4.88534069693732128461046620751, 6.28696348819327222079394110170, 6.70128422669499857385229090308, 7.70797103589107901473095663846, 8.609836530904373440437456688220, 10.02975199629696553070240282299, 10.82659253995829168080045524512, 11.46669758665371225287517015186

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