Properties

Label 1-56-56.3-r0-0-0
Degree $1$
Conductor $56$
Sign $0.386 + 0.922i$
Analytic cond. $0.260062$
Root an. cond. $0.260062$
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  + (0.5 + 0.866i)3-s + (−0.5 + 0.866i)5-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)11-s + 13-s − 15-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s − 27-s − 29-s + (−0.5 − 0.866i)31-s + (0.5 − 0.866i)33-s + (0.5 − 0.866i)37-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)3-s + (−0.5 + 0.866i)5-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)11-s + 13-s − 15-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s − 27-s − 29-s + (−0.5 − 0.866i)31-s + (0.5 − 0.866i)33-s + (0.5 − 0.866i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(56\)    =    \(2^{3} \cdot 7\)
Sign: $0.386 + 0.922i$
Analytic conductor: \(0.260062\)
Root analytic conductor: \(0.260062\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{56} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 56,\ (0:\ ),\ 0.386 + 0.922i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7831311043 + 0.5209246280i\)
\(L(\frac12)\) \(\approx\) \(0.7831311043 + 0.5209246280i\)
\(L(1)\) \(\approx\) \(0.9853529735 + 0.3949586307i\)
\(L(1)\) \(\approx\) \(0.9853529735 + 0.3949586307i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 - T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 - T \)
43 \( 1 + T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (0.5 + 0.866i)T \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 - T \)
73 \( 1 + (0.5 + 0.866i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 - T \)
89 \( 1 + (0.5 - 0.866i)T \)
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

−32.62226581537803238142745106992, −31.40354834054244039757311782718, −30.88534796706838815003140710171, −29.45440801738142322946192450052, −28.44834213985124964583053717317, −27.25379731544374331712185001285, −25.71538626442790863862629935448, −24.954033714979342472829084135097, −23.69484806656172175541247167264, −23.00688953255070787106684640005, −20.81798067798255846963686266251, −20.269692987560820180159614551004, −18.9201023277689602059161126016, −17.91090697953854904828455114495, −16.41450138176173723125088805570, −15.140616087304156406842556961449, −13.64193665242741608862021068466, −12.65598227578844444948413991532, −11.589505532440730239131325453577, −9.515195944154144649061963124194, −8.24030201410243590146529431667, −7.222322751380498019729854429015, −5.37105154696096913402046805253, −3.49915442338122875007197382990, −1.48445924299526608427941871755, 2.86592229947028444769445107760, 3.98274788399003067204061816101, 5.84367434213653341480433658441, 7.698173118292260187384135206786, 8.91619288649665427146502927957, 10.539992279648797561605690132891, 11.24474540088059881194093773060, 13.33677315490547045157807133177, 14.57460036805624808450531222602, 15.543349844025600373333509479349, 16.5811691152419186555591431359, 18.40955407094365035674089624407, 19.38365351217698016445250029531, 20.6940901781594864447242360442, 21.7569889886932054639833834159, 22.78749585396203842608140817548, 24.08505356587280379312737784967, 25.80149430434257479640600552011, 26.37753373264914409632099503886, 27.43366908793210430303059774234, 28.53313082223360704264976242599, 30.1945458082384610839988209397, 31.019623498325764102500489575255, 32.14484342853559572918264277117, 33.16506596328657197526284612113

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