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Properties

Label 2-392-56.11-c2-0-40
Degree $2$
Conductor $392$
Sign $0.629 - 0.776i$
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  + (−0.944 + 1.76i)2-s + (−1.72 + 2.98i)3-s + (−2.21 − 3.32i)4-s + (−4.22 + 2.44i)5-s + (−3.62 − 5.84i)6-s + (7.96 − 0.763i)8-s + (−1.42 − 2.46i)9-s + (−0.311 − 9.76i)10-s + (10.7 − 18.6i)11-s + (13.7 − 0.876i)12-s − 13.0i·13-s − 16.8i·15-s + (−6.17 + 14.7i)16-s + (−0.117 + 0.203i)17-s + (5.68 − 0.181i)18-s + (2.27 + 3.94i)19-s + ⋯
L(s)  = 1  + (−0.472 + 0.881i)2-s + (−0.573 + 0.993i)3-s + (−0.554 − 0.832i)4-s + (−0.845 + 0.488i)5-s + (−0.604 − 0.974i)6-s + (0.995 − 0.0954i)8-s + (−0.157 − 0.273i)9-s + (−0.0311 − 0.976i)10-s + (0.976 − 1.69i)11-s + (1.14 − 0.0730i)12-s − 1.00i·13-s − 1.12i·15-s + (−0.385 + 0.922i)16-s + (−0.00690 + 0.0119i)17-s + (0.315 − 0.0100i)18-s + (0.119 + 0.207i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.629 - 0.776i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.629 - 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $0.629 - 0.776i$
Analytic conductor: \(10.6812\)
Root analytic conductor: \(3.26821\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :1),\ 0.629 - 0.776i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.693729 + 0.330541i\)
\(L(\frac12)\) \(\approx\) \(0.693729 + 0.330541i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.944 - 1.76i)T \)
7 \( 1 \)
good3 \( 1 + (1.72 - 2.98i)T + (-4.5 - 7.79i)T^{2} \)
5 \( 1 + (4.22 - 2.44i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-10.7 + 18.6i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + 13.0iT - 169T^{2} \)
17 \( 1 + (0.117 - 0.203i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (-2.27 - 3.94i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (9.48 - 5.47i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + 34.6iT - 841T^{2} \)
31 \( 1 + (-29.5 - 17.0i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-46.9 + 27.1i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 37.8T + 1.68e3T^{2} \)
43 \( 1 + 4.84T + 1.84e3T^{2} \)
47 \( 1 + (62.6 - 36.1i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-18.7 - 10.8i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-17.4 + 30.2i)T + (-1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-55.0 + 31.8i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (9.21 - 15.9i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 47.5iT - 5.04e3T^{2} \)
73 \( 1 + (-27.9 + 48.4i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-82.2 + 47.5i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 71.5T + 6.88e3T^{2} \)
89 \( 1 + (79.8 + 138. i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 90.4T + 9.40e3T^{2} \)
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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

−11.15014874817668652409003463205, −10.25248580857614646289803152296, −9.433918854792253652110240842939, −8.308933778249309892649761142716, −7.65718371934918526750235781251, −6.27473732238977558997262151972, −5.65784020874748212068744746016, −4.38726149886468084977135153030, −3.46852441828421484268798451709, −0.58701683502079943755496186259, 0.990176941780140191936942690206, 2.07182211286507906868676027167, 3.98736123783881909647461128856, 4.68446339320470822830045982355, 6.58228773559938421791075716510, 7.26139204011684061046278626666, 8.183005824414740532658069503499, 9.261973096351497383714346085938, 9.989668226122438580947597758893, 11.50129518890354774268012645015

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