Properties

Label 2-405-5.4-c1-0-0
Degree $2$
Conductor $405$
Sign $0.977 - 0.210i$
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  − 2.52i·2-s − 4.37·4-s + (−2.18 + 0.469i)5-s + 3.46i·7-s + 5.98i·8-s + (1.18 + 5.51i)10-s + 1.37·11-s + 4.10i·13-s + 8.74·14-s + 6.37·16-s − 2.52i·17-s − 5.37·19-s + (9.55 − 2.05i)20-s − 3.46i·22-s + 5.04i·23-s + ⋯
L(s)  = 1  − 1.78i·2-s − 2.18·4-s + (−0.977 + 0.210i)5-s + 1.30i·7-s + 2.11i·8-s + (0.375 + 1.74i)10-s + 0.413·11-s + 1.13i·13-s + 2.33·14-s + 1.59·16-s − 0.612i·17-s − 1.23·19-s + (2.13 − 0.459i)20-s − 0.738i·22-s + 1.05i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.210i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.977 - 0.210i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.977 - 0.210i$
Analytic conductor: \(3.23394\)
Root analytic conductor: \(1.79831\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (244, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1/2),\ 0.977 - 0.210i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.578422 + 0.0614603i\)
\(L(\frac12)\) \(\approx\) \(0.578422 + 0.0614603i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.18 - 0.469i)T \)
good2 \( 1 + 2.52iT - 2T^{2} \)
7 \( 1 - 3.46iT - 7T^{2} \)
11 \( 1 - 1.37T + 11T^{2} \)
13 \( 1 - 4.10iT - 13T^{2} \)
17 \( 1 + 2.52iT - 17T^{2} \)
19 \( 1 + 5.37T + 19T^{2} \)
23 \( 1 - 5.04iT - 23T^{2} \)
29 \( 1 + 5.74T + 29T^{2} \)
31 \( 1 - 0.627T + 31T^{2} \)
37 \( 1 - 7.57iT - 37T^{2} \)
41 \( 1 + 1.37T + 41T^{2} \)
43 \( 1 - 3.46iT - 43T^{2} \)
47 \( 1 - 8.51iT - 47T^{2} \)
53 \( 1 + 5.34iT - 53T^{2} \)
59 \( 1 + 7.37T + 59T^{2} \)
61 \( 1 - 3.62T + 61T^{2} \)
67 \( 1 + 8.21iT - 67T^{2} \)
71 \( 1 + 4.11T + 71T^{2} \)
73 \( 1 + 7.57iT - 73T^{2} \)
79 \( 1 - 4.74T + 79T^{2} \)
83 \( 1 + 5.34iT - 83T^{2} \)
89 \( 1 - 3T + 89T^{2} \)
97 \( 1 - 18.6iT - 97T^{2} \)
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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

−11.56465905913879751176092721949, −10.71712992802168365935415107686, −9.411768804446835044596040054239, −9.037406456006516440654221527470, −7.989254645302901437089829702243, −6.48775174665028860059824798826, −4.95223863584662225306597508764, −4.01351711553007796535388758855, −2.96559672091878998730031692990, −1.81326608007659241797788030835, 0.38755697090165241954931325418, 3.81734577429622949948630999370, 4.39838057328808474185829176889, 5.63958204544276266525380409923, 6.75757879183900357649216077021, 7.41837370237657123837087413474, 8.197687797933400260973387221248, 8.849056946346994381912855998118, 10.23655963463156932497847660534, 10.99767073166108391648978202707

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