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Properties

Label 2-392-56.11-c2-0-17
Degree $2$
Conductor $392$
Sign $-0.588 - 0.808i$
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  + (−0.109 + 1.99i)2-s + (2.28 − 3.95i)3-s + (−3.97 − 0.437i)4-s + (−4.96 + 2.86i)5-s + (7.64 + 4.99i)6-s + (1.30 − 7.89i)8-s + (−5.93 − 10.2i)9-s + (−5.18 − 10.2i)10-s + (0.700 − 1.21i)11-s + (−10.8 + 14.7i)12-s + 19.0i·13-s + 26.1i·15-s + (15.6 + 3.47i)16-s + (−16.1 + 27.9i)17-s + (21.1 − 10.7i)18-s + (6.28 + 10.8i)19-s + ⋯
L(s)  = 1  + (−0.0547 + 0.998i)2-s + (0.761 − 1.31i)3-s + (−0.993 − 0.109i)4-s + (−0.992 + 0.573i)5-s + (1.27 + 0.832i)6-s + (0.163 − 0.986i)8-s + (−0.658 − 1.14i)9-s + (−0.518 − 1.02i)10-s + (0.0636 − 0.110i)11-s + (−0.900 + 1.22i)12-s + 1.46i·13-s + 1.74i·15-s + (0.976 + 0.217i)16-s + (−0.949 + 1.64i)17-s + (1.17 − 0.595i)18-s + (0.330 + 0.572i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.588 - 0.808i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.588 - 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $-0.588 - 0.808i$
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: \((2,\ 392,\ (\ :1),\ -0.588 - 0.808i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.458973 + 0.901861i\)
\(L(\frac12)\) \(\approx\) \(0.458973 + 0.901861i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.109 - 1.99i)T \)
7 \( 1 \)
good3 \( 1 + (-2.28 + 3.95i)T + (-4.5 - 7.79i)T^{2} \)
5 \( 1 + (4.96 - 2.86i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-0.700 + 1.21i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 19.0iT - 169T^{2} \)
17 \( 1 + (16.1 - 27.9i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (-6.28 - 10.8i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-13.7 + 7.94i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + 3.29iT - 841T^{2} \)
31 \( 1 + (-19.6 - 11.3i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (46.8 - 27.0i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 7.59T + 1.68e3T^{2} \)
43 \( 1 + 20.8T + 1.84e3T^{2} \)
47 \( 1 + (18.7 - 10.8i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (0.308 + 0.178i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-13.4 + 23.2i)T + (-1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (74.6 - 43.1i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (57.2 - 99.1i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 104. iT - 5.04e3T^{2} \)
73 \( 1 + (12.1 - 21.1i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-101. + 58.5i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 79.2T + 6.88e3T^{2} \)
89 \( 1 + (-1.33 - 2.30i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 52.0T + 9.40e3T^{2} \)
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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

−11.66482016514837945314787908657, −10.43357243288270539933660745974, −8.965429758928577264349819647431, −8.411692522801439063054330388863, −7.59450503374150142239476937635, −6.82326035415332208443028561768, −6.28866687374416651418837737083, −4.44284151282155094790997542118, −3.38692867824816981768735985608, −1.62430718945620080107624488268, 0.42706778319379278154165946130, 2.76029155226033128961430379692, 3.53524243957737035184426532363, 4.61363858463546273013787613643, 5.13679781544068217075482358949, 7.48905632463552365245436601801, 8.448417458348778984297022515379, 9.087979051506700443662798209063, 9.833787862882772049107481809092, 10.75474816249755130095137329067

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