↑

Properties

Label 2-392-56.5-c2-0-68
Degree $2$
Conductor $392$
Sign $-0.702 + 0.711i$
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  + (1.82 − 0.810i)2-s + (−1.16 − 2.02i)3-s + (2.68 − 2.96i)4-s + (1.55 − 2.68i)5-s + (−3.77 − 2.75i)6-s + (2.50 − 7.59i)8-s + (1.77 − 3.06i)9-s + (0.658 − 6.16i)10-s + (−4.06 + 2.34i)11-s + (−9.13 − 1.97i)12-s + 6.88·13-s − 7.24·15-s + (−1.57 − 15.9i)16-s + (−14.7 + 8.49i)17-s + (0.752 − 7.04i)18-s + (13.1 − 22.7i)19-s + ⋯
L(s)  = 1  + (0.914 − 0.405i)2-s + (−0.389 − 0.674i)3-s + (0.671 − 0.741i)4-s + (0.310 − 0.537i)5-s + (−0.629 − 0.458i)6-s + (0.313 − 0.949i)8-s + (0.196 − 0.341i)9-s + (0.0658 − 0.616i)10-s + (−0.369 + 0.213i)11-s + (−0.761 − 0.164i)12-s + 0.529·13-s − 0.482·15-s + (−0.0982 − 0.995i)16-s + (−0.865 + 0.499i)17-s + (0.0417 − 0.391i)18-s + (0.689 − 1.19i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.702 + 0.711i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.702 + 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $-0.702 + 0.711i$
Analytic conductor: \(10.6812\)
Root analytic conductor: \(3.26821\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :1),\ -0.702 + 0.711i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.01271 - 2.42332i\)
\(L(\frac12)\) \(\approx\) \(1.01271 - 2.42332i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.82 + 0.810i)T \)
7 \( 1 \)
good3 \( 1 + (1.16 + 2.02i)T + (-4.5 + 7.79i)T^{2} \)
5 \( 1 + (-1.55 + 2.68i)T + (-12.5 - 21.6i)T^{2} \)
11 \( 1 + (4.06 - 2.34i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 - 6.88T + 169T^{2} \)
17 \( 1 + (14.7 - 8.49i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-13.1 + 22.7i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (12.9 - 22.4i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + 42.2iT - 841T^{2} \)
31 \( 1 + (15.9 - 9.18i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-43.1 - 24.9i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 10.7iT - 1.68e3T^{2} \)
43 \( 1 + 24.1iT - 1.84e3T^{2} \)
47 \( 1 + (11.8 + 6.84i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-6.03 + 3.48i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-53.0 - 91.9i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-46.7 + 80.9i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (77.2 - 44.6i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 81.7T + 5.04e3T^{2} \)
73 \( 1 + (-119. + 68.9i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-6.55 + 11.3i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 2.15T + 6.88e3T^{2} \)
89 \( 1 + (-87.8 - 50.7i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 88.9iT - 9.40e3T^{2} \)
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

−11.19007357956117528315348732890, −9.949420391075246107086446161780, −9.129838724846899060855133950316, −7.65470324467237913605196292497, −6.67492495578341730404310971375, −5.86556400425570482396592411709, −4.89166062459524023606161278683, −3.71696187157626790047065416060, −2.15787189862150179650417998846, −0.923130273118708474289011634939, 2.27548234801093320361659726337, 3.59837295642091445754029279648, 4.64073726388593439218279650788, 5.58155585215055763475470831570, 6.44261699966115583315057839004, 7.48802388083925772317283157413, 8.502455376304325721548138936884, 9.872564772166657823636510328061, 10.78477571197184037640855597605, 11.25565773203647365870422909724

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