↑

Properties

Label 2-14e2-28.23-c2-0-34
Degree $2$
Conductor $196$
Sign $-0.692 + 0.721i$
Analytic cond. $5.34061$
Root an. cond. $2.31097$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.489 − 1.93i)2-s + (3.93 − 2.27i)3-s + (−3.52 − 1.89i)4-s + (0.683 − 1.18i)5-s + (−2.48 − 8.74i)6-s + (−5.40 + 5.89i)8-s + (5.82 − 10.0i)9-s + (−1.96 − 1.90i)10-s + (13.2 − 7.65i)11-s + (−18.1 + 0.529i)12-s − 19.9·13-s − 6.21i·15-s + (8.79 + 13.3i)16-s + (−2.36 − 4.10i)17-s + (−16.7 − 16.2i)18-s + (11.4 + 6.63i)19-s + ⋯
L(s)  = 1  + (0.244 − 0.969i)2-s + (1.31 − 0.757i)3-s + (−0.880 − 0.474i)4-s + (0.136 − 0.236i)5-s + (−0.413 − 1.45i)6-s + (−0.675 + 0.737i)8-s + (0.647 − 1.12i)9-s + (−0.196 − 0.190i)10-s + (1.20 − 0.695i)11-s + (−1.51 + 0.0441i)12-s − 1.53·13-s − 0.414i·15-s + (0.549 + 0.835i)16-s + (−0.139 − 0.241i)17-s + (−0.929 − 0.902i)18-s + (0.605 + 0.349i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.692 + 0.721i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.692 + 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $-0.692 + 0.721i$
Analytic conductor: \(5.34061\)
Root analytic conductor: \(2.31097\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{196} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 196,\ (\ :1),\ -0.692 + 0.721i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.921054 - 2.15972i\)
\(L(\frac12)\) \(\approx\) \(0.921054 - 2.15972i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.489 + 1.93i)T \)
7 \( 1 \)
good3 \( 1 + (-3.93 + 2.27i)T + (4.5 - 7.79i)T^{2} \)
5 \( 1 + (-0.683 + 1.18i)T + (-12.5 - 21.6i)T^{2} \)
11 \( 1 + (-13.2 + 7.65i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + 19.9T + 169T^{2} \)
17 \( 1 + (2.36 + 4.10i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (-11.4 - 6.63i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-12.9 - 7.47i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 6.20T + 841T^{2} \)
31 \( 1 + (16.0 - 9.26i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-13.7 + 23.8i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 11.1T + 1.68e3T^{2} \)
43 \( 1 - 27.0iT - 1.84e3T^{2} \)
47 \( 1 + (26.8 + 15.4i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-6.39 - 11.0i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-24.9 + 14.4i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-3.26 + 5.65i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-89.0 + 51.4i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 45.8iT - 5.04e3T^{2} \)
73 \( 1 + (35.0 + 60.7i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-28.9 - 16.6i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 159. iT - 6.88e3T^{2} \)
89 \( 1 + (25.0 - 43.3i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 89.7T + 9.40e3T^{2} \)
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

−12.13874464858307906805743669148, −11.12974559684547329211968642025, −9.558086260783553057166309608867, −9.186795981570162156813071612342, −8.082540434358293059085778887315, −6.90798648757077018955071897229, −5.23096832117244009000614985573, −3.69051768984352026688504400370, −2.62495754107741372528709234994, −1.29161015802956793776793078658, 2.71484182420210674034671325541, 4.04041910026584859634134816601, 4.92868448238896098270445733526, 6.65900253523199329312166135496, 7.54231404546257118869957031030, 8.682096355619979436930299654909, 9.457074653407549294675168606646, 10.07846364186131816188521853246, 11.90916467304021518930204666829, 12.94099477247493378264136343260

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