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Properties

Label 2-408-408.227-c0-0-1
Degree $2$
Conductor $408$
Sign $0.563 + 0.825i$
Analytic cond. $0.203618$
Root an. cond. $0.451241$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.382 − 0.923i)2-s + (0.923 + 0.382i)3-s + (−0.707 − 0.707i)4-s + (0.707 − 0.707i)6-s + (−0.923 + 0.382i)8-s + (0.707 + 0.707i)9-s + (−0.382 − 0.0761i)11-s + (−0.382 − 0.923i)12-s + i·16-s + (−0.923 − 0.382i)17-s + (0.923 − 0.382i)18-s + (−1.30 − 0.541i)19-s + (−0.216 + 0.324i)22-s − 24-s + (0.382 + 0.923i)25-s + ⋯
L(s)  = 1  + (0.382 − 0.923i)2-s + (0.923 + 0.382i)3-s + (−0.707 − 0.707i)4-s + (0.707 − 0.707i)6-s + (−0.923 + 0.382i)8-s + (0.707 + 0.707i)9-s + (−0.382 − 0.0761i)11-s + (−0.382 − 0.923i)12-s + i·16-s + (−0.923 − 0.382i)17-s + (0.923 − 0.382i)18-s + (−1.30 − 0.541i)19-s + (−0.216 + 0.324i)22-s − 24-s + (0.382 + 0.923i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.563 + 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.563 + 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(408\)    =    \(2^{3} \cdot 3 \cdot 17\)
Sign: $0.563 + 0.825i$
Analytic conductor: \(0.203618\)
Root analytic conductor: \(0.451241\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{408} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 408,\ (\ :0),\ 0.563 + 0.825i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.167030852\)
\(L(\frac12)\) \(\approx\) \(1.167030852\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.382 + 0.923i)T \)
3 \( 1 + (-0.923 - 0.382i)T \)
17 \( 1 + (0.923 + 0.382i)T \)
good5 \( 1 + (-0.382 - 0.923i)T^{2} \)
7 \( 1 + (0.382 - 0.923i)T^{2} \)
11 \( 1 + (0.382 + 0.0761i)T + (0.923 + 0.382i)T^{2} \)
13 \( 1 + iT^{2} \)
19 \( 1 + (1.30 + 0.541i)T + (0.707 + 0.707i)T^{2} \)
23 \( 1 + (0.923 + 0.382i)T^{2} \)
29 \( 1 + (0.382 + 0.923i)T^{2} \)
31 \( 1 + (0.923 - 0.382i)T^{2} \)
37 \( 1 + (-0.923 + 0.382i)T^{2} \)
41 \( 1 + (-1.63 + 1.08i)T + (0.382 - 0.923i)T^{2} \)
43 \( 1 + (1.70 - 0.707i)T + (0.707 - 0.707i)T^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + (0.707 + 0.707i)T^{2} \)
59 \( 1 + (0.707 - 0.292i)T + (0.707 - 0.707i)T^{2} \)
61 \( 1 + (-0.382 + 0.923i)T^{2} \)
67 \( 1 + 0.765iT - T^{2} \)
71 \( 1 + (0.923 - 0.382i)T^{2} \)
73 \( 1 + (-1.08 + 1.63i)T + (-0.382 - 0.923i)T^{2} \)
79 \( 1 + (0.923 + 0.382i)T^{2} \)
83 \( 1 + (-1.70 - 0.707i)T + (0.707 + 0.707i)T^{2} \)
89 \( 1 + (-0.541 + 0.541i)T - iT^{2} \)
97 \( 1 + (-0.923 - 0.617i)T + (0.382 + 0.923i)T^{2} \)
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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

−11.00433407340217327775095825659, −10.66423438098018267897514838197, −9.437381080763831838112029411767, −8.953623430415381843140096753297, −7.910900246166298277852094285835, −6.53943687973541441669127752627, −5.06198050948968082724024420023, −4.24820270240385109453908043735, −3.07685623229264143913271079430, −2.05391184680140060346904513205, 2.38491269646857317698040132903, 3.76122532379670187670661762304, 4.70964419226433234520834084455, 6.21166801826983107110790740386, 6.86458963035095939023309669031, 8.042044319975557565369528165432, 8.480421082463115911670913772440, 9.459794535966731536975946083750, 10.53166362419947117236275623006, 11.97301678649572383370574024464

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