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Properties

Label 2-405-9.4-c1-0-15
Degree $2$
Conductor $405$
Sign $-0.173 - 0.984i$
Analytic cond. $3.23394$
Root an. cond. $1.79831$
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  + (−1 − 1.73i)2-s + (−0.999 + 1.73i)4-s + (−0.5 + 0.866i)5-s + 1.99·10-s + (−2.5 − 4.33i)11-s + (−2 + 3.46i)13-s + (1.99 + 3.46i)16-s − 4·17-s − 5·19-s + (−1 − 1.73i)20-s + (−5 + 8.66i)22-s + (−3 + 5.19i)23-s + (−0.499 − 0.866i)25-s + 7.99·26-s + (2.5 + 4.33i)29-s + ⋯
L(s)  = 1  + (−0.707 − 1.22i)2-s + (−0.499 + 0.866i)4-s + (−0.223 + 0.387i)5-s + 0.632·10-s + (−0.753 − 1.30i)11-s + (−0.554 + 0.960i)13-s + (0.499 + 0.866i)16-s − 0.970·17-s − 1.14·19-s + (−0.223 − 0.387i)20-s + (−1.06 + 1.84i)22-s + (−0.625 + 1.08i)23-s + (−0.0999 − 0.173i)25-s + 1.56·26-s + (0.464 + 0.804i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.173 - 0.984i$
Analytic conductor: \(3.23394\)
Root analytic conductor: \(1.79831\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 405,\ (\ :1/2),\ -0.173 - 0.984i)\)

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 \( 1 \)
5 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (1 + 1.73i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.5 + 4.33i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2 - 3.46i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 4T + 17T^{2} \)
19 \( 1 + 5T + 19T^{2} \)
23 \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.5 - 4.33i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.5 + 7.79i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 10T + 37T^{2} \)
41 \( 1 + (3.5 - 6.06i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1 - 1.73i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1 + 1.73i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 8T + 53T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3 - 5.19i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - T + 71T^{2} \)
73 \( 1 + 8T + 73T^{2} \)
79 \( 1 + (6 + 10.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3 + 5.19i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 9T + 89T^{2} \)
97 \( 1 + (7 + 12.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.62294358370808802227290137830, −9.954365582767116809296842721188, −8.840443085737637672864224671104, −8.282841994599585323249472568018, −6.92315502455055926209526901200, −5.82378390722174223213057106595, −4.23024492524889618882193378555, −3.05208159602070366829036127000, −1.98958486613321615527244991942, 0, 2.47450709069665290588655485953, 4.44088180038416228567280094359, 5.34129194771495489156303064358, 6.57561704634863464608148049758, 7.29090938036221534532074082949, 8.265369837306651636680986082671, 8.761550421534312872876249621153, 10.02796411348234986452558275639, 10.52755959451468544356437132672

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