↑

Properties

Label 2-418-209.208-c1-0-14
Degree $2$
Conductor $418$
Sign $0.226 + 0.974i$
Analytic cond. $3.33774$
Root an. cond. $1.82695$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 1.04i·3-s + 4-s + 2.40·5-s + 1.04i·6-s − 3.19i·7-s − 8-s + 1.91·9-s − 2.40·10-s + (−2.74 − 1.86i)11-s − 1.04i·12-s + 0.0126·13-s + 3.19i·14-s − 2.50i·15-s + 16-s + 1.96i·17-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.601i·3-s + 0.5·4-s + 1.07·5-s + 0.425i·6-s − 1.20i·7-s − 0.353·8-s + 0.638·9-s − 0.761·10-s + (−0.826 − 0.562i)11-s − 0.300i·12-s + 0.00351·13-s + 0.853i·14-s − 0.647i·15-s + 0.250·16-s + 0.476i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.226 + 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.226 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(418\)    =    \(2 \cdot 11 \cdot 19\)
Sign: $0.226 + 0.974i$
Analytic conductor: \(3.33774\)
Root analytic conductor: \(1.82695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{418} (417, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 418,\ (\ :1/2),\ 0.226 + 0.974i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.931795 - 0.739948i\)
\(L(\frac12)\) \(\approx\) \(0.931795 - 0.739948i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
11 \( 1 + (2.74 + 1.86i)T \)
19 \( 1 + (-3.20 - 2.95i)T \)
good3 \( 1 + 1.04iT - 3T^{2} \)
5 \( 1 - 2.40T + 5T^{2} \)
7 \( 1 + 3.19iT - 7T^{2} \)
13 \( 1 - 0.0126T + 13T^{2} \)
17 \( 1 - 1.96iT - 17T^{2} \)
23 \( 1 + 1.11T + 23T^{2} \)
29 \( 1 + 1.28T + 29T^{2} \)
31 \( 1 + 8.41iT - 31T^{2} \)
37 \( 1 - 1.37iT - 37T^{2} \)
41 \( 1 - 5.68T + 41T^{2} \)
43 \( 1 + 4.28iT - 43T^{2} \)
47 \( 1 + 2.23T + 47T^{2} \)
53 \( 1 + 6.07iT - 53T^{2} \)
59 \( 1 - 12.6iT - 59T^{2} \)
61 \( 1 - 6.10iT - 61T^{2} \)
67 \( 1 + 2.91iT - 67T^{2} \)
71 \( 1 - 9.82iT - 71T^{2} \)
73 \( 1 + 6.87iT - 73T^{2} \)
79 \( 1 - 14.1T + 79T^{2} \)
83 \( 1 - 9.28iT - 83T^{2} \)
89 \( 1 - 6.49iT - 89T^{2} \)
97 \( 1 + 4.73iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.63843328917354565015554280186, −10.15754521396488428926292134864, −9.395946031191651515832987436659, −8.009484328008582089829290092325, −7.48463801597904893592872888956, −6.44055761968953257591102980768, −5.56406095615844171729394066217, −3.91536457535726972198681858026, −2.25528075474567765077414851654, −1.03614327660105366567684803460, 1.86191885680465912747821302604, 2.94460480807824333099001367892, 4.86539884503127954528074462858, 5.60038965185828419124840868758, 6.75760770570670892982705957201, 7.83236424840759862891547468203, 9.097274922589830831435888120915, 9.509040968391195510343969363479, 10.23209785996129248823904318885, 11.10705740134507646176751329131

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