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Properties

Label 2-405-9.4-c1-0-7
Degree $2$
Conductor $405$
Sign $0.766 - 0.642i$
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  + (0.5 + 0.866i)2-s + (0.500 − 0.866i)4-s + (−0.5 + 0.866i)5-s + 3·8-s − 0.999·10-s + (2 + 3.46i)11-s + (1 − 1.73i)13-s + (0.500 + 0.866i)16-s + 2·17-s + 4·19-s + (0.499 + 0.866i)20-s + (−1.99 + 3.46i)22-s + (−0.499 − 0.866i)25-s + 1.99·26-s + (1 + 1.73i)29-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.250 − 0.433i)4-s + (−0.223 + 0.387i)5-s + 1.06·8-s − 0.316·10-s + (0.603 + 1.04i)11-s + (0.277 − 0.480i)13-s + (0.125 + 0.216i)16-s + 0.485·17-s + 0.917·19-s + (0.111 + 0.193i)20-s + (−0.426 + 0.738i)22-s + (−0.0999 − 0.173i)25-s + 0.392·26-s + (0.185 + 0.321i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.766 - 0.642i$
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: \(0\)
Selberg data: \((2,\ 405,\ (\ :1/2),\ 0.766 - 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.75484 + 0.638712i\)
\(L(\frac12)\) \(\approx\) \(1.75484 + 0.638712i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2 - 3.46i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1 + 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1 - 1.73i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 10T + 37T^{2} \)
41 \( 1 + (5 - 8.66i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2 + 3.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4 + 6.92i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 10T + 53T^{2} \)
59 \( 1 + (-2 + 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6 - 10.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 - 10T + 73T^{2} \)
79 \( 1 + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6 + 10.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + (1 + 1.73i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.40710753686899931422878382466, −10.31207193237958031995902926787, −9.785858029022660405372615204903, −8.395278343663977067628450311541, −7.28954003786525951363732375553, −6.76524607751903835924812295554, −5.62189555635347422675265065483, −4.71759224226522898965264641231, −3.37696083523146405849527789695, −1.62793465267399250615412589107, 1.44402708365061644157728149720, 3.11613563942540129549445734355, 3.91203164759708177894860222629, 5.12960918488845465607761470672, 6.40890224773728459091900776162, 7.51564070289935064185109556564, 8.411563349310583544646181183700, 9.323640936665707545256628635492, 10.53585977360158347014900161413, 11.37781789785490517590283982447

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