↑

Properties

Label 2-392-7.5-c2-0-5
Degree $2$
Conductor $392$
Sign $-0.0633 - 0.997i$
Analytic cond. $10.6812$
Root an. cond. $3.26821$
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  + (−1.32 + 0.765i)3-s + (7.72 + 4.46i)5-s + (−3.32 + 5.76i)9-s + (4.65 + 8.06i)11-s + 0.262i·13-s − 13.6·15-s + (12.8 − 7.39i)17-s + (−22.0 − 12.7i)19-s + (−8.31 + 14.3i)23-s + (27.2 + 47.2i)25-s − 23.9i·27-s + 14.6·29-s + (−7.49 + 4.32i)31-s + (−12.3 − 7.12i)33-s + (−21.9 + 38.0i)37-s + ⋯
L(s)  = 1  + (−0.441 + 0.255i)3-s + (1.54 + 0.892i)5-s + (−0.369 + 0.640i)9-s + (0.423 + 0.733i)11-s + 0.0202i·13-s − 0.910·15-s + (0.753 − 0.434i)17-s + (−1.16 − 0.670i)19-s + (−0.361 + 0.626i)23-s + (1.09 + 1.89i)25-s − 0.887i·27-s + 0.506·29-s + (−0.241 + 0.139i)31-s + (−0.374 − 0.216i)33-s + (−0.593 + 1.02i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 - 0.997i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $-0.0633 - 0.997i$
Analytic conductor: \(10.6812\)
Root analytic conductor: \(3.26821\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (313, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :1),\ -0.0633 - 0.997i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.19080 + 1.26876i\)
\(L(\frac12)\) \(\approx\) \(1.19080 + 1.26876i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (1.32 - 0.765i)T + (4.5 - 7.79i)T^{2} \)
5 \( 1 + (-7.72 - 4.46i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-4.65 - 8.06i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 0.262iT - 169T^{2} \)
17 \( 1 + (-12.8 + 7.39i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (22.0 + 12.7i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (8.31 - 14.3i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 - 14.6T + 841T^{2} \)
31 \( 1 + (7.49 - 4.32i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (21.9 - 38.0i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 22.9iT - 1.68e3T^{2} \)
43 \( 1 + 46T + 1.84e3T^{2} \)
47 \( 1 + (-74.6 - 43.0i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-30.3 - 52.5i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-50.3 + 29.0i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (8.63 + 4.98i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-10.3 - 17.8i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 78.9T + 5.04e3T^{2} \)
73 \( 1 + (-18.5 + 10.7i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-45.1 + 78.1i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 71.8iT - 6.88e3T^{2} \)
89 \( 1 + (-1.74 - 1.00i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 158. iT - 9.40e3T^{2} \)
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

−11.11950481576918264492125305964, −10.31146451113111873857438995318, −9.823117089002071948701945805865, −8.775678559458093076823279983800, −7.36002071394407204415375480145, −6.43449035070558318805401916804, −5.64811553273638893252803480256, −4.65455458846590545875162295659, −2.89987194402689038560409887727, −1.83196655915615064565473401979, 0.819065127748438205641870471186, 2.09443913436272152777771654231, 3.83575631234885494780830681080, 5.37582536215045812456417604966, 5.93216045939465475215640547688, 6.69559467805787317346282989960, 8.434777432800262682448018490804, 8.926758974235926391207726567425, 9.977200928385888543515490005142, 10.70130305444479174710380080090

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