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Properties

Label 2-405-135.83-c1-0-12
Degree $2$
Conductor $405$
Sign $0.230 + 0.973i$
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  + (1.08 − 0.761i)2-s + (−0.0804 + 0.221i)4-s + (−1.23 − 1.86i)5-s + (0.827 − 1.77i)7-s + (0.768 + 2.86i)8-s + (−2.76 − 1.08i)10-s + (2.76 − 3.29i)11-s + (2.90 − 4.15i)13-s + (−0.451 − 2.56i)14-s + (2.66 + 2.23i)16-s + (0.828 − 3.09i)17-s + (−0.776 − 0.448i)19-s + (0.511 − 0.122i)20-s + (0.498 − 5.69i)22-s + (−5.07 + 2.36i)23-s + ⋯
L(s)  = 1  + (0.769 − 0.538i)2-s + (−0.0402 + 0.110i)4-s + (−0.551 − 0.833i)5-s + (0.312 − 0.670i)7-s + (0.271 + 1.01i)8-s + (−0.873 − 0.344i)10-s + (0.833 − 0.993i)11-s + (0.806 − 1.15i)13-s + (−0.120 − 0.684i)14-s + (0.665 + 0.558i)16-s + (0.200 − 0.749i)17-s + (−0.178 − 0.102i)19-s + (0.114 − 0.0274i)20-s + (0.106 − 1.21i)22-s + (−1.05 + 0.493i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.230 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.230 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.230 + 0.973i$
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: \((2,\ 405,\ (\ :1/2),\ 0.230 + 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49492 - 1.18242i\)
\(L(\frac12)\) \(\approx\) \(1.49492 - 1.18242i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.23 + 1.86i)T \)
good2 \( 1 + (-1.08 + 0.761i)T + (0.684 - 1.87i)T^{2} \)
7 \( 1 + (-0.827 + 1.77i)T + (-4.49 - 5.36i)T^{2} \)
11 \( 1 + (-2.76 + 3.29i)T + (-1.91 - 10.8i)T^{2} \)
13 \( 1 + (-2.90 + 4.15i)T + (-4.44 - 12.2i)T^{2} \)
17 \( 1 + (-0.828 + 3.09i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (0.776 + 0.448i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.07 - 2.36i)T + (14.7 - 17.6i)T^{2} \)
29 \( 1 + (1.14 - 6.50i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-2.27 - 0.828i)T + (23.7 + 19.9i)T^{2} \)
37 \( 1 + (-6.96 - 1.86i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (8.33 - 1.47i)T + (38.5 - 14.0i)T^{2} \)
43 \( 1 + (-0.112 - 1.28i)T + (-42.3 + 7.46i)T^{2} \)
47 \( 1 + (-7.46 - 3.48i)T + (30.2 + 36.0i)T^{2} \)
53 \( 1 + (-0.947 + 0.947i)T - 53iT^{2} \)
59 \( 1 + (-3.72 + 3.12i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (7.90 - 2.87i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (7.90 + 5.53i)T + (22.9 + 62.9i)T^{2} \)
71 \( 1 + (4.92 - 2.84i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-4.93 + 1.32i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (0.410 + 0.0724i)T + (74.2 + 27.0i)T^{2} \)
83 \( 1 + (-5.31 - 7.59i)T + (-28.3 + 77.9i)T^{2} \)
89 \( 1 + (-0.974 + 1.68i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-13.4 + 1.17i)T + (95.5 - 16.8i)T^{2} \)
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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

−11.34205341396876495804743573000, −10.52554150629318389861684618243, −9.075407352033414944350762894822, −8.270414991201254812490172372723, −7.58130293232577182872466655000, −5.96233105020058773463823567430, −4.94756606974813566703171884210, −3.95152895229891668958609516138, −3.21835077454212939836865229133, −1.14202336889060938911935856744, 1.98690384186441251970823601292, 3.89278296415643245525752993513, 4.38555618049103630733581012975, 5.95102007141987874470223110532, 6.48171477118656917783802459047, 7.44788889680243469501864914141, 8.609371436147334420719386146906, 9.703546740002308152299454804938, 10.55243614197780059064941805013, 11.75983834453548308695787738932

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