↑

Properties

Label 2-392-392.253-c1-0-37
Degree $2$
Conductor $392$
Sign $0.745 + 0.666i$
Analytic cond. $3.13013$
Root an. cond. $1.76921$
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  + (1.15 − 0.819i)2-s + (−0.0899 − 0.186i)3-s + (0.655 − 1.88i)4-s + (1.28 + 2.66i)5-s + (−0.256 − 0.141i)6-s + (−2.08 + 1.62i)7-s + (−0.793 − 2.71i)8-s + (1.84 − 2.31i)9-s + (3.66 + 2.01i)10-s + (4.48 − 3.57i)11-s + (−0.411 + 0.0475i)12-s + (1.13 − 0.908i)13-s + (−1.06 + 3.58i)14-s + (0.382 − 0.479i)15-s + (−3.14 − 2.47i)16-s + (1.58 + 6.92i)17-s + ⋯
L(s)  = 1  + (0.814 − 0.579i)2-s + (−0.0519 − 0.107i)3-s + (0.327 − 0.944i)4-s + (0.574 + 1.19i)5-s + (−0.104 − 0.0577i)6-s + (−0.788 + 0.615i)7-s + (−0.280 − 0.959i)8-s + (0.614 − 0.770i)9-s + (1.15 + 0.638i)10-s + (1.35 − 1.07i)11-s + (−0.118 + 0.0137i)12-s + (0.315 − 0.251i)13-s + (−0.285 + 0.958i)14-s + (0.0987 − 0.123i)15-s + (−0.785 − 0.619i)16-s + (0.383 + 1.67i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.745 + 0.666i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.745 + 0.666i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $0.745 + 0.666i$
Analytic conductor: \(3.13013\)
Root analytic conductor: \(1.76921\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :1/2),\ 0.745 + 0.666i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.13320 - 0.814865i\)
\(L(\frac12)\) \(\approx\) \(2.13320 - 0.814865i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.15 + 0.819i)T \)
7 \( 1 + (2.08 - 1.62i)T \)
good3 \( 1 + (0.0899 + 0.186i)T + (-1.87 + 2.34i)T^{2} \)
5 \( 1 + (-1.28 - 2.66i)T + (-3.11 + 3.90i)T^{2} \)
11 \( 1 + (-4.48 + 3.57i)T + (2.44 - 10.7i)T^{2} \)
13 \( 1 + (-1.13 + 0.908i)T + (2.89 - 12.6i)T^{2} \)
17 \( 1 + (-1.58 - 6.92i)T + (-15.3 + 7.37i)T^{2} \)
19 \( 1 + 3.79iT - 19T^{2} \)
23 \( 1 + (0.879 - 3.85i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (-1.93 + 0.442i)T + (26.1 - 12.5i)T^{2} \)
31 \( 1 + 5.81T + 31T^{2} \)
37 \( 1 + (7.94 - 1.81i)T + (33.3 - 16.0i)T^{2} \)
41 \( 1 + (8.95 - 4.31i)T + (25.5 - 32.0i)T^{2} \)
43 \( 1 + (4.28 - 8.90i)T + (-26.8 - 33.6i)T^{2} \)
47 \( 1 + (5.83 + 7.32i)T + (-10.4 + 45.8i)T^{2} \)
53 \( 1 + (-7.34 - 1.67i)T + (47.7 + 22.9i)T^{2} \)
59 \( 1 + (-1.35 + 2.82i)T + (-36.7 - 46.1i)T^{2} \)
61 \( 1 + (-3.22 + 0.736i)T + (54.9 - 26.4i)T^{2} \)
67 \( 1 + 8.46iT - 67T^{2} \)
71 \( 1 + (0.949 - 4.16i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (4.98 - 6.25i)T + (-16.2 - 71.1i)T^{2} \)
79 \( 1 - 3.23T + 79T^{2} \)
83 \( 1 + (-1.65 - 1.32i)T + (18.4 + 80.9i)T^{2} \)
89 \( 1 + (-1.09 + 1.36i)T + (-19.8 - 86.7i)T^{2} \)
97 \( 1 - 9.06T + 97T^{2} \)
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

−11.34368630760729179073042568343, −10.36452461052059092857024772094, −9.706250796228535322238941555190, −8.732823605898947237069741017736, −6.67374020746791054375202663250, −6.49775457530450229634448407420, −5.61069176445556941197125048977, −3.66519419745103836813254742900, −3.27045780452654914516326621511, −1.62623683174718004263975089390, 1.78070637780439005450956325475, 3.72276723064375766629904787363, 4.62596562416565042626016694789, 5.42221929036115766337692868995, 6.75194085350162290341065629297, 7.29701483180842397985769369541, 8.681669343817025605776710547154, 9.495046675655173733247122540639, 10.37029489415060798651656894690, 11.92242010288125152154160129122

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