↑

Properties

Label 2-418-209.5-c1-0-11
Degree $2$
Conductor $418$
Sign $0.663 + 0.748i$
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  + (−0.615 − 0.788i)2-s + (1.20 − 1.15i)3-s + (−0.241 + 0.970i)4-s + (−2.39 + 1.49i)5-s + (−1.65 − 0.232i)6-s + (2.61 + 1.16i)7-s + (0.913 − 0.406i)8-s + (−0.00748 + 0.214i)9-s + (2.65 + 0.965i)10-s + (−0.230 − 3.30i)11-s + (0.834 + 1.44i)12-s + (5.51 − 2.93i)13-s + (−0.692 − 2.77i)14-s + (−1.14 + 4.57i)15-s + (−0.882 − 0.469i)16-s + (0.0973 + 2.78i)17-s + ⋯
L(s)  = 1  + (−0.435 − 0.557i)2-s + (0.693 − 0.669i)3-s + (−0.120 + 0.485i)4-s + (−1.07 + 0.669i)5-s + (−0.674 − 0.0948i)6-s + (0.988 + 0.440i)7-s + (0.322 − 0.143i)8-s + (−0.00249 + 0.0714i)9-s + (0.839 + 0.305i)10-s + (−0.0696 − 0.997i)11-s + (0.240 + 0.417i)12-s + (1.53 − 0.813i)13-s + (−0.185 − 0.742i)14-s + (−0.294 + 1.18i)15-s + (−0.220 − 0.117i)16-s + (0.0236 + 0.676i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.663 + 0.748i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.663 + 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(418\)    =    \(2 \cdot 11 \cdot 19\)
Sign: $0.663 + 0.748i$
Analytic conductor: \(3.33774\)
Root analytic conductor: \(1.82695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{418} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 418,\ (\ :1/2),\ 0.663 + 0.748i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.23067 - 0.553362i\)
\(L(\frac12)\) \(\approx\) \(1.23067 - 0.553362i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.615 + 0.788i)T \)
11 \( 1 + (0.230 + 3.30i)T \)
19 \( 1 + (-3.77 + 2.17i)T \)
good3 \( 1 + (-1.20 + 1.15i)T + (0.104 - 2.99i)T^{2} \)
5 \( 1 + (2.39 - 1.49i)T + (2.19 - 4.49i)T^{2} \)
7 \( 1 + (-2.61 - 1.16i)T + (4.68 + 5.20i)T^{2} \)
13 \( 1 + (-5.51 + 2.93i)T + (7.26 - 10.7i)T^{2} \)
17 \( 1 + (-0.0973 - 2.78i)T + (-16.9 + 1.18i)T^{2} \)
23 \( 1 + (-6.61 + 5.54i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (4.42 - 1.26i)T + (24.5 - 15.3i)T^{2} \)
31 \( 1 + (0.618 - 0.686i)T + (-3.24 - 30.8i)T^{2} \)
37 \( 1 + (2.96 - 2.15i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (8.19 - 7.91i)T + (1.43 - 40.9i)T^{2} \)
43 \( 1 + (-3.86 - 3.23i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (3.79 + 5.61i)T + (-17.6 + 43.5i)T^{2} \)
53 \( 1 + (-8.09 - 5.05i)T + (23.2 + 47.6i)T^{2} \)
59 \( 1 + (-5.54 + 8.22i)T + (-22.1 - 54.7i)T^{2} \)
61 \( 1 + (2.68 + 6.64i)T + (-43.8 + 42.3i)T^{2} \)
67 \( 1 + (0.957 + 0.348i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (1.96 - 1.22i)T + (31.1 - 63.8i)T^{2} \)
73 \( 1 + (-0.623 - 1.27i)T + (-44.9 + 57.5i)T^{2} \)
79 \( 1 + (12.1 - 1.71i)T + (75.9 - 21.7i)T^{2} \)
83 \( 1 + (4.27 - 0.908i)T + (75.8 - 33.7i)T^{2} \)
89 \( 1 + (-2.08 - 11.8i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (0.0681 + 0.0871i)T + (-23.4 + 94.1i)T^{2} \)
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

−11.12325398165078011760811208720, −10.58196255533847166537197400940, −8.793299183591803405545493341744, −8.350116341677612243304684756529, −7.79455638232244887166555105199, −6.72799053720769732210257811341, −5.20280435507000808531387936685, −3.57768700577257468850964301853, −2.86889001727097631059790873530, −1.28119938677128984994949716352, 1.35216671095857177675412071318, 3.65851710821008473081845638837, 4.36242168538172888707830494416, 5.37495490647940467757280689526, 7.10278183067372986201112027667, 7.70294232591685017175067638167, 8.751739360932415268371111914105, 9.111322221620055906548864780488, 10.22367348232718533727730976470, 11.36055774926489792488241235186

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