Properties

Label 2-392-56.5-c2-0-3
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + 1.73i)2-s + (1.41 + 2.44i)3-s + (−1.99 − 3.46i)4-s + (−4.24 + 7.34i)5-s − 5.65·6-s + 7.99·8-s + (0.500 − 0.866i)9-s + (−8.48 − 14.6i)10-s + (5.65 − 9.79i)12-s − 25.4·13-s − 24·15-s + (−8 + 13.8i)16-s + (0.999 + 1.73i)18-s + (−4.24 + 7.34i)19-s + 33.9·20-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (0.471 + 0.816i)3-s + (−0.499 − 0.866i)4-s + (−0.848 + 1.46i)5-s − 0.942·6-s + 0.999·8-s + (0.0555 − 0.0962i)9-s + (−0.848 − 1.46i)10-s + (0.471 − 0.816i)12-s − 1.95·13-s − 1.60·15-s + (−0.5 + 0.866i)16-s + (0.0555 + 0.0962i)18-s + (−0.223 + 0.386i)19-s + 1.69·20-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: $\chi_{392} (117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :1),\ 0.0633 + 0.997i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.245684 - 0.230587i\)
\(L(\frac12)\) \(\approx\) \(0.245684 - 0.230587i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - 1.73i)T \)
7 \( 1 \)
good3 \( 1 + (-1.41 - 2.44i)T + (-4.5 + 7.79i)T^{2} \)
5 \( 1 + (4.24 - 7.34i)T + (-12.5 - 21.6i)T^{2} \)
11 \( 1 + (60.5 - 104. i)T^{2} \)
13 \( 1 + 25.4T + 169T^{2} \)
17 \( 1 + (144.5 - 250. i)T^{2} \)
19 \( 1 + (4.24 - 7.34i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-5 + 8.66i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 - 841T^{2} \)
31 \( 1 + (480.5 - 832. i)T^{2} \)
37 \( 1 + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 1.68e3T^{2} \)
43 \( 1 - 1.84e3T^{2} \)
47 \( 1 + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (38.1 + 66.1i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (4.24 - 7.34i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 110T + 5.04e3T^{2} \)
73 \( 1 + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (65 - 112. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 25.4T + 6.88e3T^{2} \)
89 \( 1 + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 9.40e3T^{2} \)
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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

−11.49654018582286181246260669828, −10.40426446369978445034932946206, −10.02496772453755613509849318905, −9.069757310802880123303722561848, −7.907613716065811827829959742803, −7.22357037660749062789783740042, −6.44074727425453942963670884192, −4.91009764036652926695445416971, −3.92391334692232488084933112347, −2.66596688348322788853646395483, 0.16626762367564172278337454049, 1.51440646260241593600473976929, 2.75789499382040349599541684444, 4.33873156400996331400541059434, 5.03602145480728480289677283295, 7.27519429898870777025891565309, 7.70798413277686910727223293120, 8.613518470112711506161324592120, 9.304043777998482741548604522564, 10.32779805642391072164161922617

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