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Properties

Label 2-14e2-28.23-c2-0-32
Degree $2$
Conductor $196$
Sign $0.827 + 0.561i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + (2.12 − 1.22i)3-s + 4·4-s + (2.82 − 4.89i)5-s + (4.24 − 2.44i)6-s + 8·8-s + (−1.50 + 2.59i)9-s + (5.65 − 9.79i)10-s + (−15 + 8.66i)11-s + (8.48 − 4.89i)12-s − 14.1·13-s − 13.8i·15-s + 16·16-s + (−9.19 − 15.9i)17-s + (−3.00 + 5.19i)18-s + (23.3 + 13.4i)19-s + ⋯
L(s)  = 1  + 2-s + (0.707 − 0.408i)3-s + 4-s + (0.565 − 0.979i)5-s + (0.707 − 0.408i)6-s + 8-s + (−0.166 + 0.288i)9-s + (0.565 − 0.979i)10-s + (−1.36 + 0.787i)11-s + (0.707 − 0.408i)12-s − 1.08·13-s − 0.923i·15-s + 16-s + (−0.540 − 0.936i)17-s + (−0.166 + 0.288i)18-s + (1.22 + 0.709i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $0.827 + 0.561i$
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: \((2,\ 196,\ (\ :1),\ 0.827 + 0.561i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.25347 - 0.999694i\)
\(L(\frac12)\) \(\approx\) \(3.25347 - 0.999694i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2T \)
7 \( 1 \)
good3 \( 1 + (-2.12 + 1.22i)T + (4.5 - 7.79i)T^{2} \)
5 \( 1 + (-2.82 + 4.89i)T + (-12.5 - 21.6i)T^{2} \)
11 \( 1 + (15 - 8.66i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + 14.1T + 169T^{2} \)
17 \( 1 + (9.19 + 15.9i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (-23.3 - 13.4i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-24 - 13.8i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 2T + 841T^{2} \)
31 \( 1 + (12.7 - 7.34i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-15 + 25.9i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 24.0T + 1.68e3T^{2} \)
43 \( 1 + 24.2iT - 1.84e3T^{2} \)
47 \( 1 + (21.2 + 12.2i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (33 + 57.1i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (57.2 - 33.0i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-22.6 + 39.1i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-6 + 3.46i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 48.4iT - 5.04e3T^{2} \)
73 \( 1 + (9.19 + 15.9i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (18 + 10.3i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 154. iT - 6.88e3T^{2} \)
89 \( 1 + (12.0 - 20.8i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 83.4T + 9.40e3T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.63627004107285740875902303125, −11.52398012317547873233454242112, −10.20088205794521373112958264061, −9.201347400517127987649695824856, −7.78389747859772534565225950181, −7.20460968959499411915929349126, −5.25263616844090790013474064425, −5.02042419411470285258243050752, −2.96516012167779148181332813266, −1.89639803616859159600974439664, 2.63013333135570448954980299367, 3.11801966993667418772710860526, 4.78605067523928798785418553713, 5.96549963199533708037282095086, 7.03306583031041237126856072784, 8.177557599644858746849183828443, 9.620020964750958727329648664232, 10.55391242312750137506232924847, 11.29362125694987090189072613277, 12.62209341859251961062720453942

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