↑

Properties

Label 2-392-56.13-c2-0-71
Degree $2$
Conductor $392$
Sign $0.332 + 0.942i$
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  + (1.28 − 1.53i)2-s + 3.89·3-s + (−0.710 − 3.93i)4-s + 8.85·5-s + (4.99 − 5.97i)6-s + (−6.95 − 3.95i)8-s + 6.18·9-s + (11.3 − 13.5i)10-s + 3.64i·11-s + (−2.76 − 15.3i)12-s − 7.79·13-s + 34.5·15-s + (−14.9 + 5.59i)16-s + 10.4i·17-s + (7.92 − 9.48i)18-s − 10.7·19-s + ⋯
L(s)  = 1  + (0.641 − 0.767i)2-s + 1.29·3-s + (−0.177 − 0.984i)4-s + 1.77·5-s + (0.832 − 0.996i)6-s + (−0.868 − 0.494i)8-s + 0.686·9-s + (1.13 − 1.35i)10-s + 0.331i·11-s + (−0.230 − 1.27i)12-s − 0.599·13-s + 2.30·15-s + (−0.936 + 0.349i)16-s + 0.616i·17-s + (0.440 − 0.527i)18-s − 0.567·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.332 + 0.942i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.332 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $0.332 + 0.942i$
Analytic conductor: \(10.6812\)
Root analytic conductor: \(3.26821\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :1),\ 0.332 + 0.942i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.47650 - 2.45939i\)
\(L(\frac12)\) \(\approx\) \(3.47650 - 2.45939i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.28 + 1.53i)T \)
7 \( 1 \)
good3 \( 1 - 3.89T + 9T^{2} \)
5 \( 1 - 8.85T + 25T^{2} \)
11 \( 1 - 3.64iT - 121T^{2} \)
13 \( 1 + 7.79T + 169T^{2} \)
17 \( 1 - 10.4iT - 289T^{2} \)
19 \( 1 + 10.7T + 361T^{2} \)
23 \( 1 + 12.9T + 529T^{2} \)
29 \( 1 - 17.2iT - 841T^{2} \)
31 \( 1 - 30.2iT - 961T^{2} \)
37 \( 1 - 39.5iT - 1.36e3T^{2} \)
41 \( 1 + 73.6iT - 1.68e3T^{2} \)
43 \( 1 + 40.8iT - 1.84e3T^{2} \)
47 \( 1 + 41.8iT - 2.20e3T^{2} \)
53 \( 1 - 6.41iT - 2.80e3T^{2} \)
59 \( 1 - 15.9T + 3.48e3T^{2} \)
61 \( 1 - 12.1T + 3.72e3T^{2} \)
67 \( 1 + 7.79iT - 4.48e3T^{2} \)
71 \( 1 + 41.3T + 5.04e3T^{2} \)
73 \( 1 + 89.6iT - 5.32e3T^{2} \)
79 \( 1 - 70.7T + 6.24e3T^{2} \)
83 \( 1 + 60.8T + 6.88e3T^{2} \)
89 \( 1 + 27.0iT - 7.92e3T^{2} \)
97 \( 1 - 3.26iT - 9.40e3T^{2} \)
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

−10.51409713292978365405211666361, −10.11946006007343538616105946617, −9.207532290464074005455217435326, −8.610894705805060356971775132527, −6.96828218613388060614062934749, −5.90485492168196142478115027779, −4.92287615537802861605238910463, −3.52421534688624006473545780970, −2.36697874348367939858990775417, −1.77822607600410703380082339143, 2.19570718688673417163607120272, 2.92761046107052605598807741187, 4.40486087415057383965603243225, 5.62764783730321581518088734670, 6.38259444868356793777848441116, 7.56248817792676116203635582561, 8.458850351595629412984774610188, 9.379037456264280475324633168416, 9.836822892843196508701617191291, 11.36010410882700857957516128266

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