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Properties

Label 2-402-201.11-c1-0-13
Degree $2$
Conductor $402$
Sign $0.960 - 0.279i$
Analytic cond. $3.20998$
Root an. cond. $1.79164$
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  + (0.327 + 0.945i)2-s + (0.637 − 1.61i)3-s + (−0.786 + 0.618i)4-s + (3.32 − 0.977i)5-s + (1.73 + 0.0760i)6-s + (−0.598 + 3.10i)7-s + (−0.841 − 0.540i)8-s + (−2.18 − 2.05i)9-s + (2.01 + 2.82i)10-s + (4.30 + 4.10i)11-s + (0.494 + 1.66i)12-s + (−3.17 + 0.151i)13-s + (−3.13 + 0.450i)14-s + (0.549 − 5.98i)15-s + (0.235 − 0.971i)16-s + (3.30 − 4.19i)17-s + ⋯
L(s)  = 1  + (0.231 + 0.668i)2-s + (0.368 − 0.929i)3-s + (−0.393 + 0.309i)4-s + (1.48 − 0.436i)5-s + (0.706 + 0.0310i)6-s + (−0.226 + 1.17i)7-s + (−0.297 − 0.191i)8-s + (−0.728 − 0.684i)9-s + (0.636 + 0.893i)10-s + (1.29 + 1.23i)11-s + (0.142 + 0.479i)12-s + (−0.881 + 0.0419i)13-s + (−0.836 + 0.120i)14-s + (0.141 − 1.54i)15-s + (0.0589 − 0.242i)16-s + (0.800 − 1.01i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 402 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 - 0.279i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 402 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.960 - 0.279i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(402\)    =    \(2 \cdot 3 \cdot 67\)
Sign: $0.960 - 0.279i$
Analytic conductor: \(3.20998\)
Root analytic conductor: \(1.79164\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{402} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 402,\ (\ :1/2),\ 0.960 - 0.279i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.01307 + 0.287514i\)
\(L(\frac12)\) \(\approx\) \(2.01307 + 0.287514i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.327 - 0.945i)T \)
3 \( 1 + (-0.637 + 1.61i)T \)
67 \( 1 + (3.05 - 7.59i)T \)
good5 \( 1 + (-3.32 + 0.977i)T + (4.20 - 2.70i)T^{2} \)
7 \( 1 + (0.598 - 3.10i)T + (-6.49 - 2.60i)T^{2} \)
11 \( 1 + (-4.30 - 4.10i)T + (0.523 + 10.9i)T^{2} \)
13 \( 1 + (3.17 - 0.151i)T + (12.9 - 1.23i)T^{2} \)
17 \( 1 + (-3.30 + 4.19i)T + (-4.00 - 16.5i)T^{2} \)
19 \( 1 + (-0.991 + 0.191i)T + (17.6 - 7.06i)T^{2} \)
23 \( 1 + (-0.781 + 8.18i)T + (-22.5 - 4.35i)T^{2} \)
29 \( 1 + (5.07 - 2.93i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.05 + 0.0980i)T + (30.8 + 2.94i)T^{2} \)
37 \( 1 + (1.31 - 2.27i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (8.91 - 3.56i)T + (29.6 - 28.2i)T^{2} \)
43 \( 1 + (-4.54 - 0.653i)T + (41.2 + 12.1i)T^{2} \)
47 \( 1 + (-0.631 - 0.450i)T + (15.3 + 44.4i)T^{2} \)
53 \( 1 + (0.651 + 4.53i)T + (-50.8 + 14.9i)T^{2} \)
59 \( 1 + (3.96 - 6.17i)T + (-24.5 - 53.6i)T^{2} \)
61 \( 1 + (6.53 + 6.85i)T + (-2.90 + 60.9i)T^{2} \)
71 \( 1 + (5.35 + 6.80i)T + (-16.7 + 68.9i)T^{2} \)
73 \( 1 + (7.47 - 7.13i)T + (3.47 - 72.9i)T^{2} \)
79 \( 1 + (0.959 - 1.86i)T + (-45.8 - 64.3i)T^{2} \)
83 \( 1 + (-13.4 - 3.26i)T + (73.7 + 38.0i)T^{2} \)
89 \( 1 + (1.86 + 0.850i)T + (58.2 + 67.2i)T^{2} \)
97 \( 1 + (10.0 + 5.83i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.89842752972006235986045059211, −9.852976618952813226168290021527, −9.296543151923176711112061054889, −8.714174347167263133906390308243, −7.31002393720584883774151382169, −6.58208681957898134641483443917, −5.72400399046061301448027661358, −4.82992305782540022546079561044, −2.79668632644525325062769648124, −1.73526508106042416364748808965, 1.65023458557533796395333434608, 3.25337468326040807427321099667, 3.89050680590534528172955336505, 5.40137617105005973627224518363, 6.12324156311612476268635168312, 7.53949812119829415381132669786, 9.053589414905196505106252715318, 9.583193922647321434144430702527, 10.33490800092999535441688286492, 10.88539387824285693282772301678

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