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Properties

Label 2-405-5.2-c2-0-30
Degree $2$
Conductor $405$
Sign $0.538 + 0.842i$
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  + (0.417 + 0.417i)2-s − 3.65i·4-s + (−0.0756 − 4.99i)5-s + (7.04 + 7.04i)7-s + (3.19 − 3.19i)8-s + (2.05 − 2.12i)10-s + 15.2·11-s + (−9.18 + 9.18i)13-s + 5.89i·14-s − 11.9·16-s + (5.30 + 5.30i)17-s − 32.8i·19-s + (−18.2 + 0.276i)20-s + (6.38 + 6.38i)22-s + (24.9 − 24.9i)23-s + ⋯
L(s)  = 1  + (0.208 + 0.208i)2-s − 0.912i·4-s + (−0.0151 − 0.999i)5-s + (1.00 + 1.00i)7-s + (0.399 − 0.399i)8-s + (0.205 − 0.212i)10-s + 1.38·11-s + (−0.706 + 0.706i)13-s + 0.420i·14-s − 0.745·16-s + (0.312 + 0.312i)17-s − 1.72i·19-s + (−0.912 + 0.0138i)20-s + (0.290 + 0.290i)22-s + (1.08 − 1.08i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.538 + 0.842i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.538 + 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.538 + 0.842i$
Analytic conductor: \(11.0354\)
Root analytic conductor: \(3.32196\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1),\ 0.538 + 0.842i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.92247 - 1.05286i\)
\(L(\frac12)\) \(\approx\) \(1.92247 - 1.05286i\)
\(L(2)\) 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 \( 1 + (0.0756 + 4.99i)T \)
good2 \( 1 + (-0.417 - 0.417i)T + 4iT^{2} \)
7 \( 1 + (-7.04 - 7.04i)T + 49iT^{2} \)
11 \( 1 - 15.2T + 121T^{2} \)
13 \( 1 + (9.18 - 9.18i)T - 169iT^{2} \)
17 \( 1 + (-5.30 - 5.30i)T + 289iT^{2} \)
19 \( 1 + 32.8iT - 361T^{2} \)
23 \( 1 + (-24.9 + 24.9i)T - 529iT^{2} \)
29 \( 1 + 36.3iT - 841T^{2} \)
31 \( 1 + 1.39T + 961T^{2} \)
37 \( 1 + (-15.9 - 15.9i)T + 1.36e3iT^{2} \)
41 \( 1 - 10.2T + 1.68e3T^{2} \)
43 \( 1 + (36.7 - 36.7i)T - 1.84e3iT^{2} \)
47 \( 1 + (8.44 + 8.44i)T + 2.20e3iT^{2} \)
53 \( 1 + (42.9 - 42.9i)T - 2.80e3iT^{2} \)
59 \( 1 - 0.115iT - 3.48e3T^{2} \)
61 \( 1 - 4.25T + 3.72e3T^{2} \)
67 \( 1 + (-42.8 - 42.8i)T + 4.48e3iT^{2} \)
71 \( 1 - 64.5T + 5.04e3T^{2} \)
73 \( 1 + (-12.7 + 12.7i)T - 5.32e3iT^{2} \)
79 \( 1 - 13.6iT - 6.24e3T^{2} \)
83 \( 1 + (22.4 - 22.4i)T - 6.88e3iT^{2} \)
89 \( 1 + 68.7iT - 7.92e3T^{2} \)
97 \( 1 + (10.6 + 10.6i)T + 9.40e3iT^{2} \)
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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

−11.21128712297265834766332199731, −9.662182568040625522619271895872, −9.134698741331404489700104471352, −8.367739428139324780213634212419, −6.93475566712549351338377087286, −6.01037299040161018741684223498, −4.84828437503201136384518134759, −4.52595340960428101399184002456, −2.19825934509886808581872747117, −1.01884897897469706079676568379, 1.62318013914638854887966657450, 3.28211000699753886301087426875, 3.92880116819579210305625945478, 5.21104725575593051625226204361, 6.77859909074132287527898389121, 7.48038139435931807798602033262, 8.122591224548711623667368070538, 9.473945400770952514240128525730, 10.52013192409393182438115341177, 11.26164219996965416437898752209

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