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Properties

Label 2-405-135.2-c1-0-9
Degree $2$
Conductor $405$
Sign $0.999 + 0.00445i$
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  + (0.761 + 1.08i)2-s + (0.0804 − 0.221i)4-s + (−1.62 − 1.53i)5-s + (1.77 + 0.827i)7-s + (2.86 − 0.768i)8-s + (0.439 − 2.93i)10-s + (2.76 − 3.29i)11-s + (−4.15 − 2.90i)13-s + (0.451 + 2.56i)14-s + (2.66 + 2.23i)16-s + (3.09 + 0.828i)17-s + (0.776 + 0.448i)19-s + (−0.470 + 0.234i)20-s + (5.69 + 0.498i)22-s + (2.36 + 5.07i)23-s + ⋯
L(s)  = 1  + (0.538 + 0.769i)2-s + (0.0402 − 0.110i)4-s + (−0.725 − 0.688i)5-s + (0.670 + 0.312i)7-s + (1.01 − 0.271i)8-s + (0.138 − 0.929i)10-s + (0.833 − 0.993i)11-s + (−1.15 − 0.806i)13-s + (0.120 + 0.684i)14-s + (0.665 + 0.558i)16-s + (0.749 + 0.200i)17-s + (0.178 + 0.102i)19-s + (−0.105 + 0.0524i)20-s + (1.21 + 0.106i)22-s + (0.493 + 1.05i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00445i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00445i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.999 + 0.00445i$
Analytic conductor: \(3.23394\)
Root analytic conductor: \(1.79831\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (332, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1/2),\ 0.999 + 0.00445i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.87696 - 0.00418324i\)
\(L(\frac12)\) \(\approx\) \(1.87696 - 0.00418324i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.62 + 1.53i)T \)
good2 \( 1 + (-0.761 - 1.08i)T + (-0.684 + 1.87i)T^{2} \)
7 \( 1 + (-1.77 - 0.827i)T + (4.49 + 5.36i)T^{2} \)
11 \( 1 + (-2.76 + 3.29i)T + (-1.91 - 10.8i)T^{2} \)
13 \( 1 + (4.15 + 2.90i)T + (4.44 + 12.2i)T^{2} \)
17 \( 1 + (-3.09 - 0.828i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-0.776 - 0.448i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.36 - 5.07i)T + (-14.7 + 17.6i)T^{2} \)
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 \( 1 + (1.86 - 6.96i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (8.33 - 1.47i)T + (38.5 - 14.0i)T^{2} \)
43 \( 1 + (-1.28 + 0.112i)T + (42.3 - 7.46i)T^{2} \)
47 \( 1 + (3.48 - 7.46i)T + (-30.2 - 36.0i)T^{2} \)
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 \( 1 + (-5.53 + 7.90i)T + (-22.9 - 62.9i)T^{2} \)
71 \( 1 + (4.92 - 2.84i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.32 + 4.93i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-0.410 - 0.0724i)T + (74.2 + 27.0i)T^{2} \)
83 \( 1 + (-7.59 + 5.31i)T + (28.3 - 77.9i)T^{2} \)
89 \( 1 + (0.974 - 1.68i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.17 - 13.4i)T + (-95.5 + 16.8i)T^{2} \)
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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

−11.54812970173310553772284414026, −10.36768517913466118441216993842, −9.319335350853141991279872662819, −8.092374649482704118046105120101, −7.67676455735521877808469043392, −6.37461469461234889465027636517, −5.34735213761991456879672162445, −4.73166757576866744645879874538, −3.41396595991809683721517098788, −1.24516452207141572610305073045, 1.86211776254636970875108603205, 3.11998956543857143126901484553, 4.26024883097949169159790530827, 4.88102876215266760976867260808, 6.96302690645508525184199104311, 7.26157704633109886693900851470, 8.400372602369753322866756277169, 9.749878549960816889307692499035, 10.64015050818526952059538490858, 11.42057406254611170158856548322

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