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Properties

Label 2-402-201.11-c1-0-18
Degree $2$
Conductor $402$
Sign $0.561 + 0.827i$
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 + (0.0461 − 1.73i)3-s + (−0.786 + 0.618i)4-s + (1.51 − 0.443i)5-s + (1.65 − 0.522i)6-s + (0.543 − 2.82i)7-s + (−0.841 − 0.540i)8-s + (−2.99 − 0.159i)9-s + (0.913 + 1.28i)10-s + (−3.41 − 3.25i)11-s + (1.03 + 1.38i)12-s + (3.72 − 0.177i)13-s + (2.84 − 0.409i)14-s + (−0.698 − 2.63i)15-s + (0.235 − 0.971i)16-s + (−2.27 + 2.89i)17-s + ⋯
L(s)  = 1  + (0.231 + 0.668i)2-s + (0.0266 − 0.999i)3-s + (−0.393 + 0.309i)4-s + (0.676 − 0.198i)5-s + (0.674 − 0.213i)6-s + (0.205 − 1.06i)7-s + (−0.297 − 0.191i)8-s + (−0.998 − 0.0532i)9-s + (0.289 + 0.405i)10-s + (−1.03 − 0.982i)11-s + (0.298 + 0.401i)12-s + (1.03 − 0.0491i)13-s + (0.760 − 0.109i)14-s + (−0.180 − 0.681i)15-s + (0.0589 − 0.242i)16-s + (−0.552 + 0.702i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 402 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.561 + 0.827i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 402 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.561 + 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(402\)    =    \(2 \cdot 3 \cdot 67\)
Sign: $0.561 + 0.827i$
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: \((2,\ 402,\ (\ :1/2),\ 0.561 + 0.827i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34966 - 0.715150i\)
\(L(\frac12)\) \(\approx\) \(1.34966 - 0.715150i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0461 + 1.73i)T \)
67 \( 1 + (7.47 + 3.33i)T \)
good5 \( 1 + (-1.51 + 0.443i)T + (4.20 - 2.70i)T^{2} \)
7 \( 1 + (-0.543 + 2.82i)T + (-6.49 - 2.60i)T^{2} \)
11 \( 1 + (3.41 + 3.25i)T + (0.523 + 10.9i)T^{2} \)
13 \( 1 + (-3.72 + 0.177i)T + (12.9 - 1.23i)T^{2} \)
17 \( 1 + (2.27 - 2.89i)T + (-4.00 - 16.5i)T^{2} \)
19 \( 1 + (-1.80 + 0.347i)T + (17.6 - 7.06i)T^{2} \)
23 \( 1 + (-0.483 + 5.05i)T + (-22.5 - 4.35i)T^{2} \)
29 \( 1 + (-1.69 + 0.980i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-10.7 - 0.511i)T + (30.8 + 2.94i)T^{2} \)
37 \( 1 + (-2.29 + 3.98i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.92 + 2.37i)T + (29.6 - 28.2i)T^{2} \)
43 \( 1 + (8.85 + 1.27i)T + (41.2 + 12.1i)T^{2} \)
47 \( 1 + (-7.78 - 5.54i)T + (15.3 + 44.4i)T^{2} \)
53 \( 1 + (-1.57 - 10.9i)T + (-50.8 + 14.9i)T^{2} \)
59 \( 1 + (6.62 - 10.3i)T + (-24.5 - 53.6i)T^{2} \)
61 \( 1 + (-1.20 - 1.26i)T + (-2.90 + 60.9i)T^{2} \)
71 \( 1 + (0.536 + 0.682i)T + (-16.7 + 68.9i)T^{2} \)
73 \( 1 + (9.21 - 8.78i)T + (3.47 - 72.9i)T^{2} \)
79 \( 1 + (1.97 - 3.83i)T + (-45.8 - 64.3i)T^{2} \)
83 \( 1 + (-12.1 - 2.94i)T + (73.7 + 38.0i)T^{2} \)
89 \( 1 + (5.80 + 2.64i)T + (58.2 + 67.2i)T^{2} \)
97 \( 1 + (-6.58 - 3.80i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.04473322911863987178661059534, −10.39221540080696692335577357873, −8.875351728972376764404619346431, −8.198556944510360762122427912026, −7.38629624108780395668407482832, −6.25308181531226234398395290631, −5.75430625186959089570215302610, −4.31807345186998766744495706571, −2.80513008232010907534218284079, −0.984334312978346064130478416152, 2.17311534990057325483078622473, 3.11327916261373638607261603367, 4.60812610227558221544605335148, 5.35035100926071606715772227431, 6.27266884246040345021219408438, 8.047359851408961793472010811795, 9.002429807152648960078230310295, 9.805492396483266011175981073287, 10.35159192656737395735623943159, 11.45098263184708976803887714037

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