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Properties

Label 1-3968-3968.1083-r0-0-0
Degree $1$
Conductor $3968$
Sign $0.991 - 0.130i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3968 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3968 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3968\)    =    \(2^{7} \cdot 31\)
Sign: $0.991 - 0.130i$
Analytic conductor: \(18.4273\)
Root analytic conductor: \(18.4273\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3968} (1083, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3968,\ (0:\ ),\ 0.991 - 0.130i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8740890055 - 0.05729478347i\)
\(L(\frac12)\) \(\approx\) \(0.8740890055 - 0.05729478347i\)
\(L(1)\) \(\approx\) \(0.7058533488 + 0.1350925938i\)
\(L(1)\) \(\approx\) \(0.7058533488 + 0.1350925938i\)

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

−18.466778829524647769046099087, −17.86890736642684202686635238861, −17.19132691516029100774187937472, −16.50996072099150436371764397697, −15.91917322443138775511532715546, −15.20469352173956822436869746383, −14.399523536913218313221478596832, −13.62031872843400459779468068746, −12.87512557671394259821958952533, −12.35270218398755584076266288597, −11.68019699985121170682432884791, −10.88813686071859547152236940290, −10.44125595853970477159965953682, −9.84170910922448575966019846442, −8.33481325092317565657875667347, −7.74345039727168827594701189568, −7.50946780719915261243208831768, −6.6988350790378226445644254195, −5.76423538925843506951657787106, −4.96018572353952550944121166319, −4.46818004425836963331021525295, −3.305182446916316630967003121196, −2.75587061266438756626309992336, −1.293968585977512449364867731320, −0.78344833065333273709301805387, 0.425708100881771747791651577068, 1.47896521998971434412837034355, 2.85292986300540871104881723913, 3.3545575554684909268113002135, 4.40407979197616090866903961112, 5.085811599803207563567108297777, 5.43854657124280482540910429317, 6.46575212334706565354980578466, 7.26268248879461450030129251565, 8.02349693038474118449724801766, 8.876447627226729738416972267904, 9.38848362692251931296073241497, 10.28147770682567263877811801466, 11.137784966300301384523290835853, 11.54471350377107212716117199401, 12.15992049410260033245158864337, 12.68867618252661130702985501332, 13.73569266622481676834348891697, 14.69239377196331450375595187102, 15.23591324610042503330274402179, 15.85098116829783368428638606281, 16.37883440224777767322365571235, 16.83952926910145870140946944138, 18.07185948167355530955608289491, 18.2658550695748787508723911038

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