Properties

Label 2-405-135.83-c1-0-3
Degree $2$
Conductor $405$
Sign $0.387 - 0.921i$
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  + (1.22 − 0.856i)2-s + (0.0790 − 0.217i)4-s + (−2.10 + 0.751i)5-s + (−2.03 + 4.36i)7-s + (0.683 + 2.55i)8-s + (−1.93 + 2.72i)10-s + (2.25 − 2.68i)11-s + (−2.18 + 3.11i)13-s + (1.24 + 7.08i)14-s + (3.37 + 2.83i)16-s + (0.367 − 1.37i)17-s + (1.30 + 0.750i)19-s + (−0.00328 + 0.517i)20-s + (0.455 − 5.21i)22-s + (−1.35 + 0.633i)23-s + ⋯
L(s)  = 1  + (0.865 − 0.605i)2-s + (0.0395 − 0.108i)4-s + (−0.941 + 0.336i)5-s + (−0.769 + 1.65i)7-s + (0.241 + 0.902i)8-s + (−0.611 + 0.861i)10-s + (0.678 − 0.808i)11-s + (−0.605 + 0.864i)13-s + (0.333 + 1.89i)14-s + (0.844 + 0.708i)16-s + (0.0891 − 0.332i)17-s + (0.298 + 0.172i)19-s + (−0.000733 + 0.115i)20-s + (0.0971 − 1.11i)22-s + (−0.283 + 0.132i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.387 - 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.387 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.387 - 0.921i$
Analytic conductor: \(3.23394\)
Root analytic conductor: \(1.79831\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1/2),\ 0.387 - 0.921i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22309 + 0.812736i\)
\(L(\frac12)\) \(\approx\) \(1.22309 + 0.812736i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.10 - 0.751i)T \)
good2 \( 1 + (-1.22 + 0.856i)T + (0.684 - 1.87i)T^{2} \)
7 \( 1 + (2.03 - 4.36i)T + (-4.49 - 5.36i)T^{2} \)
11 \( 1 + (-2.25 + 2.68i)T + (-1.91 - 10.8i)T^{2} \)
13 \( 1 + (2.18 - 3.11i)T + (-4.44 - 12.2i)T^{2} \)
17 \( 1 + (-0.367 + 1.37i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.30 - 0.750i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.35 - 0.633i)T + (14.7 - 17.6i)T^{2} \)
29 \( 1 + (-0.168 + 0.957i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (2.44 + 0.891i)T + (23.7 + 19.9i)T^{2} \)
37 \( 1 + (-6.69 - 1.79i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-0.670 + 0.118i)T + (38.5 - 14.0i)T^{2} \)
43 \( 1 + (0.0175 + 0.200i)T + (-42.3 + 7.46i)T^{2} \)
47 \( 1 + (-7.89 - 3.68i)T + (30.2 + 36.0i)T^{2} \)
53 \( 1 + (2.81 - 2.81i)T - 53iT^{2} \)
59 \( 1 + (5.69 - 4.77i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (1.08 - 0.396i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-12.5 - 8.79i)T + (22.9 + 62.9i)T^{2} \)
71 \( 1 + (-11.1 + 6.42i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (0.343 - 0.0920i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-9.66 - 1.70i)T + (74.2 + 27.0i)T^{2} \)
83 \( 1 + (6.51 + 9.30i)T + (-28.3 + 77.9i)T^{2} \)
89 \( 1 + (2.09 - 3.63i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.41 + 0.561i)T + (95.5 - 16.8i)T^{2} \)
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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

−11.77634949237087169474363420572, −11.02617996530267894865307876794, −9.513368552855250711194179844686, −8.781043472240937627163796849220, −7.78285837728443059622076771551, −6.47380843262574971369420640383, −5.51886469229684840178833381389, −4.28716593513607995012902271514, −3.29151180120175862885977858190, −2.43844860885888964601096415526, 0.75022747580311212708759917191, 3.52542698148374402631186320841, 4.16432322838963367412448289431, 5.06245763319967509543852978873, 6.46104592381420848285295890719, 7.22141755773599173949333113491, 7.82008889399151131980949153194, 9.451557855126933846683664953995, 10.14752851046835814630827583984, 11.07395326233726167116919003799

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