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Properties

Label 2-405-15.14-c2-0-21
Degree $2$
Conductor $405$
Sign $0.380 - 0.924i$
Analytic cond. $11.0354$
Root an. cond. $3.32196$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.29·2-s + 1.25·4-s + (4.62 + 1.90i)5-s + 6.98i·7-s − 6.28·8-s + (10.6 + 4.36i)10-s + 18.2i·11-s − 4.23i·13-s + 16.0i·14-s − 19.4·16-s + 17.3·17-s − 3.96·19-s + (5.82 + 2.39i)20-s + 41.8i·22-s + 0.575·23-s + ⋯
L(s)  = 1  + 1.14·2-s + 0.314·4-s + (0.924 + 0.380i)5-s + 0.997i·7-s − 0.785·8-s + (1.06 + 0.436i)10-s + 1.65i·11-s − 0.325i·13-s + 1.14i·14-s − 1.21·16-s + 1.01·17-s − 0.208·19-s + (0.291 + 0.119i)20-s + 1.90i·22-s + 0.0250·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.380 - 0.924i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.380 - 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.380 - 0.924i$
Analytic conductor: \(11.0354\)
Root analytic conductor: \(3.32196\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (404, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1),\ 0.380 - 0.924i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.54648 + 1.70505i\)
\(L(\frac12)\) \(\approx\) \(2.54648 + 1.70505i\)
\(L(2)\) 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 \( 1 + (-4.62 - 1.90i)T \)
good2 \( 1 - 2.29T + 4T^{2} \)
7 \( 1 - 6.98iT - 49T^{2} \)
11 \( 1 - 18.2iT - 121T^{2} \)
13 \( 1 + 4.23iT - 169T^{2} \)
17 \( 1 - 17.3T + 289T^{2} \)
19 \( 1 + 3.96T + 361T^{2} \)
23 \( 1 - 0.575T + 529T^{2} \)
29 \( 1 + 20.9iT - 841T^{2} \)
31 \( 1 - 33.4T + 961T^{2} \)
37 \( 1 - 21.4iT - 1.36e3T^{2} \)
41 \( 1 + 50.9iT - 1.68e3T^{2} \)
43 \( 1 - 8.26iT - 1.84e3T^{2} \)
47 \( 1 + 15.1T + 2.20e3T^{2} \)
53 \( 1 + 24.5T + 2.80e3T^{2} \)
59 \( 1 + 49.8iT - 3.48e3T^{2} \)
61 \( 1 - 62.9T + 3.72e3T^{2} \)
67 \( 1 - 119. iT - 4.48e3T^{2} \)
71 \( 1 + 66.8iT - 5.04e3T^{2} \)
73 \( 1 + 48.9iT - 5.32e3T^{2} \)
79 \( 1 + 117.T + 6.24e3T^{2} \)
83 \( 1 - 7.32T + 6.88e3T^{2} \)
89 \( 1 + 100. iT - 7.92e3T^{2} \)
97 \( 1 - 4.14iT - 9.40e3T^{2} \)
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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

−11.58859293538037006536734110983, −10.11869575256026408225017471880, −9.633790909565473140260513618675, −8.544985942196573703666129891933, −7.14707705317999965940838604992, −6.10726326310693815403415057326, −5.40388878119961021866005847930, −4.50059343255823549462758746253, −3.03992867705049721104143035319, −2.07496965258192849472744562943, 0.972512086218730084365022212655, 2.91853056324144622018053625213, 3.92465859099147988601831306479, 5.04405850471478336384933397103, 5.88097330729257809759874902644, 6.66257089285976365199929411640, 8.147298155099862693066412020126, 9.069179663871288489510112405801, 10.05865099322843250602282220209, 10.99218664394220480024605146881

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