↑

Properties

Label 2-218-109.45-c1-0-3
Degree $2$
Conductor $218$
Sign $0.237 - 0.971i$
Analytic cond. $1.74073$
Root an. cond. $1.31937$
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 + (−0.176 + 0.305i)3-s + 4-s + (−2.04 + 3.54i)5-s + (−0.176 + 0.305i)6-s + (−1.41 + 2.45i)7-s + 8-s + (1.43 + 2.48i)9-s + (−2.04 + 3.54i)10-s + (−2.46 − 4.26i)11-s + (−0.176 + 0.305i)12-s + (2.43 − 4.22i)13-s + (−1.41 + 2.45i)14-s + (−0.723 − 1.25i)15-s + 16-s + 6.02·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.101 + 0.176i)3-s + 0.5·4-s + (−0.915 + 1.58i)5-s + (−0.0721 + 0.124i)6-s + (−0.535 + 0.927i)7-s + 0.353·8-s + (0.479 + 0.829i)9-s + (−0.647 + 1.12i)10-s + (−0.743 − 1.28i)11-s + (−0.0509 + 0.0883i)12-s + (0.676 − 1.17i)13-s + (−0.378 + 0.656i)14-s + (−0.186 − 0.323i)15-s + 0.250·16-s + 1.46·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 218 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.237 - 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 218 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.237 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(218\)    =    \(2 \cdot 109\)
Sign: $0.237 - 0.971i$
Analytic conductor: \(1.74073\)
Root analytic conductor: \(1.31937\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{218} (45, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 218,\ (\ :1/2),\ 0.237 - 0.971i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17468 + 0.921765i\)
\(L(\frac12)\) \(\approx\) \(1.17468 + 0.921765i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
109 \( 1 + (10.0 + 2.80i)T \)
good3 \( 1 + (0.176 - 0.305i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (2.04 - 3.54i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (1.41 - 2.45i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.46 + 4.26i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.43 + 4.22i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 6.02T + 17T^{2} \)
19 \( 1 - 5.63T + 19T^{2} \)
23 \( 1 + 2.13T + 23T^{2} \)
29 \( 1 + (2.28 - 3.96i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.896 - 1.55i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.96 - 5.13i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 3.14T + 41T^{2} \)
43 \( 1 - 3.50T + 43T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.414 + 0.717i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.90 + 6.75i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.81 + 6.61i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.76 + 13.4i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 6.66T + 71T^{2} \)
73 \( 1 + (-4.41 - 7.64i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.29 + 7.43i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.67 + 6.37i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (3.54 - 6.14i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-8.01 + 13.8i)T + (-48.5 - 84.0i)T^{2} \)
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

−12.45167173293942538506435024522, −11.48308410356232404748351409622, −10.75502240754827561904871032924, −10.04925984043131001864063908679, −8.105643614644483992104372105450, −7.52911856614908312548293601851, −6.11975431841465219605379980093, −5.35661827205954447902715318701, −3.33863070245610257369751588416, −3.04042118557611992509922700783, 1.17666522203094515272580049456, 3.78812568732877869504449036729, 4.37531711774672089038528535559, 5.64293503238778818846538856997, 7.18387979331821679969838409705, 7.74417601475023153552687467435, 9.281536637748267316353481342986, 10.05993875661466866230291005621, 11.71703061161688869688068394009, 12.15170567533084065948421742878

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