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Properties

Label 2-405-45.22-c2-0-17
Degree $2$
Conductor $405$
Sign $-0.0438 - 0.999i$
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  + (−0.0822 + 0.307i)2-s + (3.37 + 1.94i)4-s + (−0.799 + 4.93i)5-s + (1.26 − 4.71i)7-s + (−1.77 + 1.77i)8-s + (−1.44 − 0.651i)10-s + (5.67 + 9.82i)11-s + (−2.03 − 7.58i)13-s + (1.34 + 0.775i)14-s + (7.39 + 12.8i)16-s + (17.3 + 17.3i)17-s + 8.69i·19-s + (−12.3 + 15.1i)20-s + (−3.48 + 0.933i)22-s + (−4.22 − 15.7i)23-s + ⋯
L(s)  = 1  + (−0.0411 + 0.153i)2-s + (0.844 + 0.487i)4-s + (−0.159 + 0.987i)5-s + (0.180 − 0.673i)7-s + (−0.221 + 0.221i)8-s + (−0.144 − 0.0651i)10-s + (0.515 + 0.893i)11-s + (−0.156 − 0.583i)13-s + (0.0959 + 0.0553i)14-s + (0.462 + 0.800i)16-s + (1.02 + 1.02i)17-s + 0.457i·19-s + (−0.616 + 0.755i)20-s + (−0.158 + 0.0424i)22-s + (−0.183 − 0.686i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0438 - 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0438 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.0438 - 0.999i$
Analytic conductor: \(11.0354\)
Root analytic conductor: \(3.32196\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (352, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1),\ -0.0438 - 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.34714 + 1.40759i\)
\(L(\frac12)\) \(\approx\) \(1.34714 + 1.40759i\)
\(L(2)\) 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 \( 1 + (0.799 - 4.93i)T \)
good2 \( 1 + (0.0822 - 0.307i)T + (-3.46 - 2i)T^{2} \)
7 \( 1 + (-1.26 + 4.71i)T + (-42.4 - 24.5i)T^{2} \)
11 \( 1 + (-5.67 - 9.82i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (2.03 + 7.58i)T + (-146. + 84.5i)T^{2} \)
17 \( 1 + (-17.3 - 17.3i)T + 289iT^{2} \)
19 \( 1 - 8.69iT - 361T^{2} \)
23 \( 1 + (4.22 + 15.7i)T + (-458. + 264.5i)T^{2} \)
29 \( 1 + (30.4 - 17.5i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (5.34 - 9.26i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (6.04 + 6.04i)T + 1.36e3iT^{2} \)
41 \( 1 + (-0.348 + 0.603i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-36.1 - 9.69i)T + (1.60e3 + 924.5i)T^{2} \)
47 \( 1 + (16.1 - 60.4i)T + (-1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (-0.696 + 0.696i)T - 2.80e3iT^{2} \)
59 \( 1 + (34.5 + 19.9i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (2.95 + 5.11i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-61.6 + 16.5i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 - 68T + 5.04e3T^{2} \)
73 \( 1 + (-77.7 + 77.7i)T - 5.32e3iT^{2} \)
79 \( 1 + (21.2 - 12.2i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-17.9 - 4.81i)T + (5.96e3 + 3.44e3i)T^{2} \)
89 \( 1 + 82.1iT - 7.92e3T^{2} \)
97 \( 1 + (9.00 - 33.5i)T + (-8.14e3 - 4.70e3i)T^{2} \)
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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

−11.03468690466872449667811747453, −10.60975559007741552313325541565, −9.667972289752886653034631354023, −8.099320195490699922611043919732, −7.51871790542895642892143161139, −6.74839797227792917312113547691, −5.80735697327148284157045204956, −4.07827957232333465102523424583, −3.17776018278723831087369912068, −1.78949499960953838761540798862, 0.876558492795030873646339710801, 2.20375233442273358734375031459, 3.66528562859628321501787529449, 5.21987539538330244746190332733, 5.81076190982879469687117919051, 7.05139331222509846501731154624, 8.076978448810839943852229836810, 9.185782318898760528062044217991, 9.680165702034531935807202173284, 11.08546693423563838244461370103

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