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Properties

Label 2-392-56.45-c2-0-2
Degree $2$
Conductor $392$
Sign $-0.706 - 0.708i$
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  + (−0.215 − 1.98i)2-s + (−0.455 + 0.788i)3-s + (−3.90 + 0.856i)4-s + (3.17 + 5.49i)5-s + (1.66 + 0.735i)6-s + (2.54 + 7.58i)8-s + (4.08 + 7.07i)9-s + (10.2 − 7.49i)10-s + (−11.4 − 6.60i)11-s + (1.10 − 3.47i)12-s − 19.4·13-s − 5.77·15-s + (14.5 − 6.69i)16-s + (−13.7 − 7.96i)17-s + (13.1 − 9.64i)18-s + (−8.22 − 14.2i)19-s + ⋯
L(s)  = 1  + (−0.107 − 0.994i)2-s + (−0.151 + 0.262i)3-s + (−0.976 + 0.214i)4-s + (0.634 + 1.09i)5-s + (0.277 + 0.122i)6-s + (0.318 + 0.948i)8-s + (0.453 + 0.786i)9-s + (1.02 − 0.749i)10-s + (−1.04 − 0.600i)11-s + (0.0919 − 0.289i)12-s − 1.49·13-s − 0.385·15-s + (0.908 − 0.418i)16-s + (−0.811 − 0.468i)17-s + (0.732 − 0.535i)18-s + (−0.433 − 0.750i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.706 - 0.708i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.706 - 0.708i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $-0.706 - 0.708i$
Analytic conductor: \(10.6812\)
Root analytic conductor: \(3.26821\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (325, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :1),\ -0.706 - 0.708i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.110381 + 0.265957i\)
\(L(\frac12)\) \(\approx\) \(0.110381 + 0.265957i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.215 + 1.98i)T \)
7 \( 1 \)
good3 \( 1 + (0.455 - 0.788i)T + (-4.5 - 7.79i)T^{2} \)
5 \( 1 + (-3.17 - 5.49i)T + (-12.5 + 21.6i)T^{2} \)
11 \( 1 + (11.4 + 6.60i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + 19.4T + 169T^{2} \)
17 \( 1 + (13.7 + 7.96i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (8.22 + 14.2i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (11.9 + 20.7i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 - 16.6iT - 841T^{2} \)
31 \( 1 + (-11.1 - 6.42i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (41.1 - 23.7i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 6.49iT - 1.68e3T^{2} \)
43 \( 1 - 33.2iT - 1.84e3T^{2} \)
47 \( 1 + (-18.9 + 10.9i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-32.2 - 18.5i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (27.3 - 47.3i)T + (-1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (5.12 + 8.87i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-14.8 - 8.56i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 32.0T + 5.04e3T^{2} \)
73 \( 1 + (92.8 + 53.5i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-29.1 - 50.4i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 36.3T + 6.88e3T^{2} \)
89 \( 1 + (0.929 - 0.536i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 169. iT - 9.40e3T^{2} \)
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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

−11.10360341593777417404879914196, −10.44524728470774994799911255265, −10.11623137813763253076089303101, −8.963876777776195134024189485083, −7.80295109421493227772357380191, −6.78351425479738172776107972884, −5.30343114752395596135921278566, −4.52993533564399475867318684154, −2.81144231000045927785756392927, −2.28334656825790562083190727924, 0.12483332833013398197908245938, 1.85981740930720130407100720787, 4.15035772380281295306404041349, 5.09293752021040411265472601176, 5.88215489255518023561264437204, 7.02308141478754135236089803768, 7.83423941596416344060526920287, 8.869052453487388743505145847273, 9.707929245560517780412775720112, 10.24702461597356725840222427251

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