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Properties

Label 1-392-392.101-r1-0-0
Degree $1$
Conductor $392$
Sign $-0.695 - 0.718i$
Analytic cond. $42.1262$
Root an. cond. $42.1262$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.988 − 0.149i)3-s + (0.365 + 0.930i)5-s + (0.955 + 0.294i)9-s + (−0.955 + 0.294i)11-s + (−0.222 + 0.974i)13-s + (−0.222 − 0.974i)15-s + (−0.826 − 0.563i)17-s + (−0.5 + 0.866i)19-s + (0.826 − 0.563i)23-s + (−0.733 + 0.680i)25-s + (−0.900 − 0.433i)27-s + (0.900 − 0.433i)29-s + (0.5 + 0.866i)31-s + (0.988 − 0.149i)33-s + (−0.0747 + 0.997i)37-s + ⋯
L(s)  = 1  + (−0.988 − 0.149i)3-s + (0.365 + 0.930i)5-s + (0.955 + 0.294i)9-s + (−0.955 + 0.294i)11-s + (−0.222 + 0.974i)13-s + (−0.222 − 0.974i)15-s + (−0.826 − 0.563i)17-s + (−0.5 + 0.866i)19-s + (0.826 − 0.563i)23-s + (−0.733 + 0.680i)25-s + (−0.900 − 0.433i)27-s + (0.900 − 0.433i)29-s + (0.5 + 0.866i)31-s + (0.988 − 0.149i)33-s + (−0.0747 + 0.997i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.695 - 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.695 - 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $-0.695 - 0.718i$
Analytic conductor: \(42.1262\)
Root analytic conductor: \(42.1262\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 392,\ (1:\ ),\ -0.695 - 0.718i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02774539583 + 0.06549373723i\)
\(L(\frac12)\) \(\approx\) \(0.02774539583 + 0.06549373723i\)
\(L(1)\) \(\approx\) \(0.6469492780 + 0.1539712289i\)
\(L(1)\) \(\approx\) \(0.6469492780 + 0.1539712289i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.988 - 0.149i)T \)
5 \( 1 + (0.365 + 0.930i)T \)
11 \( 1 + (-0.955 + 0.294i)T \)
13 \( 1 + (-0.222 + 0.974i)T \)
17 \( 1 + (-0.826 - 0.563i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (0.826 - 0.563i)T \)
29 \( 1 + (0.900 - 0.433i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (-0.0747 + 0.997i)T \)
41 \( 1 + (-0.623 + 0.781i)T \)
43 \( 1 + (-0.623 - 0.781i)T \)
47 \( 1 + (0.733 + 0.680i)T \)
53 \( 1 + (-0.0747 - 0.997i)T \)
59 \( 1 + (0.365 - 0.930i)T \)
61 \( 1 + (0.0747 - 0.997i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (-0.900 - 0.433i)T \)
73 \( 1 + (0.733 - 0.680i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (-0.222 - 0.974i)T \)
89 \( 1 + (-0.955 - 0.294i)T \)
97 \( 1 - T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.71826937315941417154400430116, −22.91779173117641474134611586229, −21.78618146005022844109854628445, −21.312918488012704420741022691247, −20.32585608834884925499153422058, −19.3461876709084877892829410653, −18.08623235696158125932899823363, −17.479694145707306482322992074948, −16.79187164613309097146417671913, −15.72898987140556510738826538926, −15.23729997998312499787771076861, −13.418286450412546903753613785913, −12.99439282859875439468069808489, −12.094412757779680353290005202392, −10.92904441836964828183405486150, −10.29661914090310495493943072271, −9.15896615550508607752767044938, −8.16546505013669000131097964223, −6.93432398165973668204009157036, −5.7534340288117997345418117429, −5.14245242063829257118124731551, −4.202958691142830035331935443936, −2.53810976903600906030649326432, −1.03670193765923581517078623778, −0.02524066911542265018021073637, 1.71927512667726978830176550130, 2.794886703960291388123623282136, 4.39156289607925889206281947487, 5.273648079854285291374050845867, 6.573300964446960122139526133409, 6.879679840936931506831849802803, 8.21329524946214403680448813623, 9.72890438040001291528464064558, 10.43210168659292407732894819925, 11.234604972440697772389861805222, 12.124768742042890457768267310002, 13.168815109414837809312716532747, 14.03925014488593047622715359007, 15.16261298502290316809424372577, 16.00164367615699632521505486543, 17.019415216891945303094191866178, 17.78736590099536232851886868670, 18.61099229135351605251863320324, 19.11831329947147512480688199543, 20.68294592187731353498937365191, 21.49745834405988694996227733330, 22.219695276539409987564249319623, 23.107790448086883470760966716770, 23.60944698111220783412889240502, 24.75236584235980527608803131890

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