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Properties

Label 1-3968-3968.1101-r0-0-0
Degree $1$
Conductor $3968$
Sign $0.730 + 0.682i$
Analytic cond. $18.4273$
Root an. cond. $18.4273$
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  + (0.785 + 0.619i)3-s + (0.980 − 0.195i)5-s + (−0.233 + 0.972i)7-s + (0.233 + 0.972i)9-s + (0.938 − 0.346i)11-s + (−0.117 − 0.993i)13-s + (0.891 + 0.453i)15-s + (−0.987 − 0.156i)17-s + (0.872 + 0.488i)19-s + (−0.785 + 0.619i)21-s + (0.972 − 0.233i)23-s + (0.923 − 0.382i)25-s + (−0.418 + 0.908i)27-s + (0.962 − 0.271i)29-s + (0.951 + 0.309i)33-s + ⋯
L(s)  = 1  + (0.785 + 0.619i)3-s + (0.980 − 0.195i)5-s + (−0.233 + 0.972i)7-s + (0.233 + 0.972i)9-s + (0.938 − 0.346i)11-s + (−0.117 − 0.993i)13-s + (0.891 + 0.453i)15-s + (−0.987 − 0.156i)17-s + (0.872 + 0.488i)19-s + (−0.785 + 0.619i)21-s + (0.972 − 0.233i)23-s + (0.923 − 0.382i)25-s + (−0.418 + 0.908i)27-s + (0.962 − 0.271i)29-s + (0.951 + 0.309i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3968 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.730 + 0.682i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3968 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.730 + 0.682i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3968\)    =    \(2^{7} \cdot 31\)
Sign: $0.730 + 0.682i$
Analytic conductor: \(18.4273\)
Root analytic conductor: \(18.4273\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3968} (1101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3968,\ (0:\ ),\ 0.730 + 0.682i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.081215080 + 1.214858616i\)
\(L(\frac12)\) \(\approx\) \(3.081215080 + 1.214858616i\)
\(L(1)\) \(\approx\) \(1.728047149 + 0.4229290697i\)
\(L(1)\) \(\approx\) \(1.728047149 + 0.4229290697i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
31 \( 1 \)
good3 \( 1 + (0.785 + 0.619i)T \)
5 \( 1 + (0.980 - 0.195i)T \)
7 \( 1 + (-0.233 + 0.972i)T \)
11 \( 1 + (0.938 - 0.346i)T \)
13 \( 1 + (-0.117 - 0.993i)T \)
17 \( 1 + (-0.987 - 0.156i)T \)
19 \( 1 + (0.872 + 0.488i)T \)
23 \( 1 + (0.972 - 0.233i)T \)
29 \( 1 + (0.962 - 0.271i)T \)
37 \( 1 + (0.195 + 0.980i)T \)
41 \( 1 + (0.649 - 0.760i)T \)
43 \( 1 + (0.785 - 0.619i)T \)
47 \( 1 + (0.453 - 0.891i)T \)
53 \( 1 + (0.0392 + 0.999i)T \)
59 \( 1 + (-0.488 - 0.872i)T \)
61 \( 1 + (-0.831 + 0.555i)T \)
67 \( 1 + (-0.831 + 0.555i)T \)
71 \( 1 + (-0.852 - 0.522i)T \)
73 \( 1 + (0.852 - 0.522i)T \)
79 \( 1 + (0.156 - 0.987i)T \)
83 \( 1 + (-0.872 - 0.488i)T \)
89 \( 1 + (0.972 + 0.233i)T \)
97 \( 1 + (-0.587 + 0.809i)T \)
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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

−18.31348369393740564740530466964, −17.72679129673217165057336162875, −17.268991935666906549975826640601, −16.50713375996243786217710166264, −15.645511990780295353810730802256, −14.67146899682594454382348602581, −14.18529259144186347880434677066, −13.76235359717549965743435968269, −13.088298365081340863961577929836, −12.50848331335399472228102309844, −11.49303079693104508163865414599, −10.858714337472952698891585573310, −9.86794857887583589644010729272, −9.19435149491503539253566357357, −9.045770006017011577995786817534, −7.77927469024377965796870659613, −6.96953946892164335031615597050, −6.74166460105783009954157415275, −5.9889880280431316854213134890, −4.68063497061270408268885239606, −4.127600067135844180844894133534, −3.12166688781289200627516171584, −2.46204693548466765801181215179, −1.511689409291416802489985008845, −1.021708579375771645808033067588, 1.02289761814974277378768696653, 2.0085772075352938363783348546, 2.773725972881422549293101246298, 3.24314971222520684569362416238, 4.36938075397214222460545379897, 5.10234306203692448259752355107, 5.78546082919432352984393752552, 6.470841778971326117176536872305, 7.46890788965379610600843770881, 8.42982781776041773401267510774, 9.05586620394409066577612073492, 9.29198096144923888110221347539, 10.22109998610055545496802609643, 10.72931363092038020798971090292, 11.785371528606016317853567390588, 12.485865816170228598067229455542, 13.32396382663027428886235505061, 13.794457528691018162948499914392, 14.517806566661011739185616440812, 15.19552983106410678592912924528, 15.72865814068569559398858795988, 16.481463676021876491926796692357, 17.214572415670318587058231129161, 17.87295658338076181060220008052, 18.65334501941538059299206617783

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