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Properties

Label 2-406-7.2-c1-0-4
Degree $2$
Conductor $406$
Sign $0.970 - 0.239i$
Analytic cond. $3.24192$
Root an. cond. $1.80053$
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  + (−0.5 + 0.866i)2-s + (−1.61 − 2.80i)3-s + (−0.499 − 0.866i)4-s + (−0.572 + 0.991i)5-s + 3.23·6-s + (0.469 + 2.60i)7-s + 0.999·8-s + (−3.73 + 6.46i)9-s + (−0.572 − 0.991i)10-s + (0.425 + 0.736i)11-s + (−1.61 + 2.80i)12-s + 3.66·13-s + (−2.48 − 0.895i)14-s + 3.70·15-s + (−0.5 + 0.866i)16-s + (−0.0971 − 0.168i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.933 − 1.61i)3-s + (−0.249 − 0.433i)4-s + (−0.255 + 0.443i)5-s + 1.32·6-s + (0.177 + 0.984i)7-s + 0.353·8-s + (−1.24 + 2.15i)9-s + (−0.180 − 0.313i)10-s + (0.128 + 0.222i)11-s + (−0.466 + 0.808i)12-s + 1.01·13-s + (−0.665 − 0.239i)14-s + 0.955·15-s + (−0.125 + 0.216i)16-s + (−0.0235 − 0.0408i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 406 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.239i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 406 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 - 0.239i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(406\)    =    \(2 \cdot 7 \cdot 29\)
Sign: $0.970 - 0.239i$
Analytic conductor: \(3.24192\)
Root analytic conductor: \(1.80053\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{406} (233, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 406,\ (\ :1/2),\ 0.970 - 0.239i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.777734 + 0.0944357i\)
\(L(\frac12)\) \(\approx\) \(0.777734 + 0.0944357i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-0.469 - 2.60i)T \)
29 \( 1 + T \)
good3 \( 1 + (1.61 + 2.80i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.572 - 0.991i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.425 - 0.736i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 3.66T + 13T^{2} \)
17 \( 1 + (0.0971 + 0.168i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.49 + 6.05i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.73 + 2.99i)T + (-11.5 - 19.9i)T^{2} \)
31 \( 1 + (-1.98 - 3.44i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.60 - 6.24i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 4.04T + 41T^{2} \)
43 \( 1 - 9.77T + 43T^{2} \)
47 \( 1 + (0.0269 - 0.0466i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.33 - 2.31i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.02 - 6.96i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.54 + 4.40i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.80 - 10.0i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 11.2T + 71T^{2} \)
73 \( 1 + (-3.65 - 6.33i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.67 + 11.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 9.04T + 83T^{2} \)
89 \( 1 + (7.11 - 12.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 18.7T + 97T^{2} \)
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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

−11.35728899210989105627566526535, −10.74187247024516940673574667487, −9.100794787017297194182954510050, −8.315276816009515401296647328322, −7.29524943579801342615447358423, −6.70823315345895789782052651889, −5.85498246569107409336890758711, −4.99819449311258280821761452212, −2.65987415526625686169579561304, −1.13178229940640331611444409566, 0.851976642188361233399330373561, 3.60425973440226220241552597735, 4.03677650483768899987299587188, 5.15537939946194177167017749331, 6.17692850348891812900497723862, 7.75558261345216626846731518240, 8.854264027063859698605163552300, 9.658350447534555451731085342971, 10.45547820212573166495554127702, 11.01828025524186905272067392081

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