↑

Properties

Label 2-14e2-28.3-c1-0-10
Degree $2$
Conductor $196$
Sign $0.983 + 0.182i$
Analytic cond. $1.56506$
Root an. cond. $1.25102$
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  + (−0.0345 + 1.41i)2-s + (1.36 − 2.35i)3-s + (−1.99 − 0.0976i)4-s + (0.937 − 0.541i)5-s + (3.28 + 2.00i)6-s + (0.207 − 2.82i)8-s + (−2.20 − 3.82i)9-s + (0.732 + 1.34i)10-s + (1.80 + 1.04i)11-s + (−2.94 + 4.57i)12-s − 2.61i·13-s − 2.94i·15-s + (3.98 + 0.390i)16-s + (3.86 + 2.23i)17-s + (5.48 − 2.98i)18-s + (−0.563 − 0.976i)19-s + ⋯
L(s)  = 1  + (−0.0244 + 0.999i)2-s + (0.786 − 1.36i)3-s + (−0.998 − 0.0488i)4-s + (0.419 − 0.242i)5-s + (1.34 + 0.819i)6-s + (0.0732 − 0.997i)8-s + (−0.735 − 1.27i)9-s + (0.231 + 0.424i)10-s + (0.544 + 0.314i)11-s + (−0.851 + 1.32i)12-s − 0.724i·13-s − 0.760i·15-s + (0.995 + 0.0975i)16-s + (0.936 + 0.540i)17-s + (1.29 − 0.704i)18-s + (−0.129 − 0.224i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.182i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 + 0.182i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $0.983 + 0.182i$
Analytic conductor: \(1.56506\)
Root analytic conductor: \(1.25102\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{196} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 196,\ (\ :1/2),\ 0.983 + 0.182i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41606 - 0.130239i\)
\(L(\frac12)\) \(\approx\) \(1.41606 - 0.130239i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0345 - 1.41i)T \)
7 \( 1 \)
good3 \( 1 + (-1.36 + 2.35i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.937 + 0.541i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.80 - 1.04i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.61iT - 13T^{2} \)
17 \( 1 + (-3.86 - 2.23i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.563 + 0.976i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (6.16 - 3.55i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 1.17T + 29T^{2} \)
31 \( 1 + (3.85 - 6.66i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2 - 3.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 5.54iT - 41T^{2} \)
43 \( 1 - 7.97iT - 43T^{2} \)
47 \( 1 + (2.72 + 4.71i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.24 + 5.61i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.41 + 7.64i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (11.3 - 6.53i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.71 - 5.03i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (6.90 + 3.98i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.60 + 2.08i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 4.31T + 83T^{2} \)
89 \( 1 + (-3.47 + 2.00i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 3.82iT - 97T^{2} \)
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

−12.85481627054739808542112979684, −12.01321364744068584741915909573, −10.10041249145281963723086749713, −9.109786428019371788735074158331, −8.135370590782040776887834990186, −7.47365337758152360592905349277, −6.43549297205570923058867411801, −5.42591411680003620395547266459, −3.54625919730159844872921886800, −1.53103318350388944199827379551, 2.32555184230429772886839164630, 3.64201961658231058503831776189, 4.44428785294912028866287586632, 5.87818660493049044166263304980, 7.961405617325695195952155204310, 9.031363546304414806818660367155, 9.688755756451976158311907615685, 10.34473234183547876190570158422, 11.35714876959298706871808234103, 12.33489414012113828758174539159

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