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Properties

Label 2-406-7.4-c1-0-13
Degree $2$
Conductor $406$
Sign $0.388 + 0.921i$
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 + (0.357 − 0.618i)3-s + (−0.499 + 0.866i)4-s + (−0.116 − 0.201i)5-s − 0.714·6-s + (2.36 − 1.18i)7-s + 0.999·8-s + (1.24 + 2.15i)9-s + (−0.116 + 0.201i)10-s + (−0.146 + 0.254i)11-s + (0.357 + 0.618i)12-s + 4.15·13-s + (−2.20 − 1.46i)14-s − 0.166·15-s + (−0.5 − 0.866i)16-s + (−0.304 + 0.527i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.206 − 0.357i)3-s + (−0.249 + 0.433i)4-s + (−0.0521 − 0.0903i)5-s − 0.291·6-s + (0.895 − 0.446i)7-s + 0.353·8-s + (0.414 + 0.718i)9-s + (−0.0368 + 0.0638i)10-s + (−0.0442 + 0.0766i)11-s + (0.103 + 0.178i)12-s + 1.15·13-s + (−0.589 − 0.390i)14-s − 0.0430·15-s + (−0.125 − 0.216i)16-s + (−0.0739 + 0.128i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 406 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.388 + 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 406 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.388 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(406\)    =    \(2 \cdot 7 \cdot 29\)
Sign: $0.388 + 0.921i$
Analytic conductor: \(3.24192\)
Root analytic conductor: \(1.80053\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{406} (291, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 406,\ (\ :1/2),\ 0.388 + 0.921i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13922 - 0.756063i\)
\(L(\frac12)\) \(\approx\) \(1.13922 - 0.756063i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.36 + 1.18i)T \)
29 \( 1 + T \)
good3 \( 1 + (-0.357 + 0.618i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.116 + 0.201i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.146 - 0.254i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 4.15T + 13T^{2} \)
17 \( 1 + (0.304 - 0.527i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.33 + 5.78i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.743 - 1.28i)T + (-11.5 + 19.9i)T^{2} \)
31 \( 1 + (-0.675 + 1.17i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.318 + 0.551i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 3.31T + 41T^{2} \)
43 \( 1 + 3.29T + 43T^{2} \)
47 \( 1 + (1.09 + 1.88i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.99 - 10.3i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.15 - 3.72i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.82 + 3.15i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.47 - 4.29i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 4.21T + 71T^{2} \)
73 \( 1 + (2.92 - 5.06i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.05 + 1.82i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6.73T + 83T^{2} \)
89 \( 1 + (-6.31 - 10.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 5.70T + 97T^{2} \)
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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

−10.85987647149880202210481612117, −10.56378714661948079524162256078, −9.145262381424350730050775878239, −8.349737467684860798507691211942, −7.63164946960974963735796425965, −6.56526684008597122279075944734, −4.95273921214531411653873540416, −4.08132886613802234936508539041, −2.48049727077483476197382697780, −1.24732823064923001899545718496, 1.53685395416606056083215366510, 3.51202186549112144867401273535, 4.63354341520505666180651493411, 5.81558351333485792829006503125, 6.68824340652993213507695146781, 7.930884721893287507358036615481, 8.617718976772753060436022903489, 9.389196993528842809794565080666, 10.45766613819347558352394626417, 11.20181238004617537394157703314

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