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Properties

Label 2-392-56.19-c1-0-21
Degree $2$
Conductor $392$
Sign $-0.248 + 0.968i$
Analytic cond. $3.13013$
Root an. cond. $1.76921$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.10 − 0.878i)2-s + (−0.662 + 0.382i)3-s + (0.455 + 1.94i)4-s + (1.51 − 2.62i)5-s + (1.07 + 0.158i)6-s + (1.20 − 2.55i)8-s + (−1.20 + 2.09i)9-s + (−3.98 + 1.57i)10-s + (−2.12 − 3.67i)11-s + (−1.04 − 1.11i)12-s + 3.02·13-s + 2.31i·15-s + (−3.58 + 1.77i)16-s + (3.86 − 2.23i)17-s + (3.17 − 1.25i)18-s + (2.92 + 1.68i)19-s + ⋯
L(s)  = 1  + (−0.783 − 0.621i)2-s + (−0.382 + 0.220i)3-s + (0.227 + 0.973i)4-s + (0.677 − 1.17i)5-s + (0.437 + 0.0647i)6-s + (0.426 − 0.904i)8-s + (−0.402 + 0.696i)9-s + (−1.25 + 0.498i)10-s + (−0.639 − 1.10i)11-s + (−0.302 − 0.322i)12-s + 0.839·13-s + 0.598i·15-s + (−0.896 + 0.443i)16-s + (0.936 − 0.540i)17-s + (0.748 − 0.295i)18-s + (0.671 + 0.387i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.248 + 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.248 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $-0.248 + 0.968i$
Analytic conductor: \(3.13013\)
Root analytic conductor: \(1.76921\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :1/2),\ -0.248 + 0.968i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.502092 - 0.647258i\)
\(L(\frac12)\) \(\approx\) \(0.502092 - 0.647258i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.10 + 0.878i)T \)
7 \( 1 \)
good3 \( 1 + (0.662 - 0.382i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.51 + 2.62i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.12 + 3.67i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 3.02T + 13T^{2} \)
17 \( 1 + (-3.86 + 2.23i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.92 - 1.68i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.84 + 2.79i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 5.59iT - 29T^{2} \)
31 \( 1 + (5.16 + 8.95i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.00 - 1.15i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 7.07iT - 41T^{2} \)
43 \( 1 - 2.58T + 43T^{2} \)
47 \( 1 + (3.02 - 5.24i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.83 - 1.63i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-9.32 + 5.38i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.65 + 6.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1 - 1.73i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 14.4iT - 71T^{2} \)
73 \( 1 + (2.92 - 1.68i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-9.69 - 5.59i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 9.23iT - 83T^{2} \)
89 \( 1 + (-2.14 - 1.24i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 6.88iT - 97T^{2} \)
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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

−11.04332085126566163514294096180, −10.06011072885835100925912783442, −9.367833268769561881583704980234, −8.296809299020802079804037172868, −7.87710803469161299400312785141, −5.98954307271718522191415746570, −5.31983743771691539102049148633, −3.87103995058276997700136577878, −2.34421209401780578655645578634, −0.76385031869659278416026318594, 1.65608303212273335123879941200, 3.22742176376032711163280035871, 5.28176035418384452308491569752, 6.07814142873292251221608541680, 6.87904672430621925358781459070, 7.63047172420711662459934320869, 8.866268741118275744866129380348, 9.865428032844088466481784568676, 10.43436290831621813997749909311, 11.26376010328238243871889133052

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