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Properties

Label 2-392-56.11-c2-0-13
Degree $2$
Conductor $392$
Sign $-0.162 - 0.986i$
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  + (1.26 − 1.54i)2-s + (−1.70 + 2.95i)3-s + (−0.791 − 3.92i)4-s + (−1.34 + 0.774i)5-s + (2.41 + 6.38i)6-s + (−7.07 − 3.74i)8-s + (−1.32 − 2.30i)9-s + (−0.500 + 3.05i)10-s + (2.24 − 3.88i)11-s + (12.9 + 4.35i)12-s + 1.54i·13-s − 5.29i·15-s + (−14.7 + 6.20i)16-s + (−11.8 + 20.4i)17-s + (−5.24 − 0.858i)18-s + (12.4 + 21.5i)19-s + ⋯
L(s)  = 1  + (0.633 − 0.773i)2-s + (−0.569 + 0.985i)3-s + (−0.197 − 0.980i)4-s + (−0.268 + 0.154i)5-s + (0.402 + 1.06i)6-s + (−0.883 − 0.467i)8-s + (−0.147 − 0.255i)9-s + (−0.0500 + 0.305i)10-s + (0.203 − 0.353i)11-s + (1.07 + 0.362i)12-s + 0.119i·13-s − 0.352i·15-s + (−0.921 + 0.387i)16-s + (−0.695 + 1.20i)17-s + (−0.291 − 0.0476i)18-s + (0.654 + 1.13i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.162 - 0.986i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.162 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $-0.162 - 0.986i$
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: \((2,\ 392,\ (\ :1),\ -0.162 - 0.986i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.614267 + 0.724020i\)
\(L(\frac12)\) \(\approx\) \(0.614267 + 0.724020i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.26 + 1.54i)T \)
7 \( 1 \)
good3 \( 1 + (1.70 - 2.95i)T + (-4.5 - 7.79i)T^{2} \)
5 \( 1 + (1.34 - 0.774i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-2.24 + 3.88i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 1.54iT - 169T^{2} \)
17 \( 1 + (11.8 - 20.4i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (-12.4 - 21.5i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (30.5 - 17.6i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 - 22.4iT - 841T^{2} \)
31 \( 1 + (40.4 + 23.3i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-50.7 + 29.2i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 26.9T + 1.68e3T^{2} \)
43 \( 1 + 17.1T + 1.84e3T^{2} \)
47 \( 1 + (-31.2 + 18.0i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-84.7 - 48.9i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (30.7 - 53.3i)T + (-1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-32.6 + 18.8i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-16.6 + 28.9i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 102. iT - 5.04e3T^{2} \)
73 \( 1 + (34.6 - 60.0i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (33.5 - 19.3i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 3.61T + 6.88e3T^{2} \)
89 \( 1 + (22.0 + 38.1i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 96.1T + 9.40e3T^{2} \)
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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

−11.29840136930399580956876139560, −10.59206175770083683372143601581, −9.885224313060535252075325022703, −9.002062252099564361576397386756, −7.59022783617259844411855125105, −6.00216690381950678611765956661, −5.47641069900073161728817922359, −4.08360476411437487153847259271, −3.68866583163519922173914636842, −1.81644228363234411998547913021, 0.35199952416737476868592594143, 2.44962755747733803772544114420, 4.08524217481693482543230831404, 5.11183698307198829943052544424, 6.22192568085664521580111523065, 6.95519037772193399582571926950, 7.64533649712313571989757872182, 8.692024323052803079388773240389, 9.783341947857556608827526963079, 11.48563277458578549287685809572

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