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Properties

Label 2-392-56.45-c2-0-20
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: $\chi_{392} (325, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :1),\ 0.0633 - 0.997i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.943669 + 0.885682i\)
\(L(\frac12)\) \(\approx\) \(0.943669 + 0.885682i\)
\(L(2)\) 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

−11.01044074308298678807570227963, −10.49540407752141406404872106566, −9.873408089924168135208749021219, −8.932258518060387074046115876601, −7.70488946817470115297522595834, −6.50561855663540992989160781856, −5.50101705135615644796188014257, −3.99235288356955229742321431178, −3.11653345360547991509085994362, −1.69607128792641034346872345437, 0.817049253821624342113664848996, 1.54219579443392315373091080358, 4.26156110973976428261130940309, 5.51005653435672483105474544738, 6.05706233022637177551826580116, 6.95844419856996022186915442009, 8.241285167964172904174769591807, 8.851184956401896085305551050991, 9.616186441978912152212166511943, 10.75883590484889852458823180721

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