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Properties

Label 2-405-135.113-c1-0-6
Degree $2$
Conductor $405$
Sign $0.661 - 0.749i$
Analytic cond. $3.23394$
Root an. cond. $1.79831$
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  + (1.75 + 0.153i)2-s + (1.09 + 0.192i)4-s + (−0.131 + 2.23i)5-s + (2.75 + 1.93i)7-s + (−1.51 − 0.406i)8-s + (−0.574 + 3.90i)10-s + (1.11 − 3.07i)11-s + (0.506 + 5.79i)13-s + (4.55 + 3.81i)14-s + (−4.68 − 1.70i)16-s + (−1.11 + 0.298i)17-s + (6.40 − 3.69i)19-s + (−0.574 + 2.41i)20-s + (2.43 − 5.22i)22-s + (−1.09 − 1.56i)23-s + ⋯
L(s)  = 1  + (1.24 + 0.108i)2-s + (0.546 + 0.0963i)4-s + (−0.0589 + 0.998i)5-s + (1.04 + 0.730i)7-s + (−0.536 − 0.143i)8-s + (−0.181 + 1.23i)10-s + (0.337 − 0.926i)11-s + (0.140 + 1.60i)13-s + (1.21 + 1.02i)14-s + (−1.17 − 0.426i)16-s + (−0.270 + 0.0723i)17-s + (1.46 − 0.848i)19-s + (−0.128 + 0.539i)20-s + (0.519 − 1.11i)22-s + (−0.228 − 0.326i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.661 - 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.661 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.661 - 0.749i$
Analytic conductor: \(3.23394\)
Root analytic conductor: \(1.79831\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (368, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1/2),\ 0.661 - 0.749i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.31876 + 1.04594i\)
\(L(\frac12)\) \(\approx\) \(2.31876 + 1.04594i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.131 - 2.23i)T \)
good2 \( 1 + (-1.75 - 0.153i)T + (1.96 + 0.347i)T^{2} \)
7 \( 1 + (-2.75 - 1.93i)T + (2.39 + 6.57i)T^{2} \)
11 \( 1 + (-1.11 + 3.07i)T + (-8.42 - 7.07i)T^{2} \)
13 \( 1 + (-0.506 - 5.79i)T + (-12.8 + 2.25i)T^{2} \)
17 \( 1 + (1.11 - 0.298i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-6.40 + 3.69i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.09 + 1.56i)T + (-7.86 + 21.6i)T^{2} \)
29 \( 1 + (0.381 - 0.320i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (-0.573 + 3.25i)T + (-29.1 - 10.6i)T^{2} \)
37 \( 1 + (1.17 + 4.40i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-1.39 + 1.65i)T + (-7.11 - 40.3i)T^{2} \)
43 \( 1 + (1.68 + 3.62i)T + (-27.6 + 32.9i)T^{2} \)
47 \( 1 + (-2.84 + 4.06i)T + (-16.0 - 44.1i)T^{2} \)
53 \( 1 + (1.25 - 1.25i)T - 53iT^{2} \)
59 \( 1 + (9.92 - 3.61i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (1.82 + 10.3i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (5.82 - 0.509i)T + (65.9 - 11.6i)T^{2} \)
71 \( 1 + (-6.62 - 3.82i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (1.84 - 6.89i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (5.08 + 6.06i)T + (-13.7 + 77.7i)T^{2} \)
83 \( 1 + (0.544 - 6.22i)T + (-81.7 - 14.4i)T^{2} \)
89 \( 1 + (0.260 + 0.450i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.27 + 1.99i)T + (62.3 - 74.3i)T^{2} \)
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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

−11.57455441857903791242667040405, −11.00517951844344468690094341519, −9.429424574994996490960431485363, −8.701114473301028340481378651401, −7.33058089994516699452207617925, −6.40015585773929344348778538824, −5.55583862958619847086134520463, −4.49952582422594696422455171024, −3.45411219325537656934369572854, −2.24513555346016639307555197584, 1.39771573173073518143430264979, 3.28825124946405011987807333188, 4.42442705663954219803646441158, 5.03639535596888555197577488797, 5.89237922156561239578954832472, 7.48848156223931838315687239782, 8.164335497050766775088849737196, 9.357812327616767518658636664110, 10.37002857007492655467384267886, 11.53228856927173238571282814374

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