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Properties

Label 1-31e2-961.9-r0-0-0
Degree $1$
Conductor $961$
Sign $0.996 + 0.0857i$
Analytic cond. $4.46286$
Root an. cond. $4.46286$
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  + (−0.546 + 0.837i)2-s + (0.999 + 0.0135i)3-s + (−0.403 − 0.914i)4-s + (0.943 − 0.331i)5-s + (−0.557 + 0.830i)6-s + (−0.860 + 0.508i)7-s + (0.986 + 0.161i)8-s + (0.999 + 0.0270i)9-s + (−0.237 + 0.971i)10-s + (−0.340 − 0.940i)11-s + (−0.391 − 0.920i)12-s + (0.648 − 0.760i)13-s + (0.0439 − 0.999i)14-s + (0.947 − 0.318i)15-s + (−0.674 + 0.738i)16-s + (−0.921 − 0.388i)17-s + ⋯
L(s)  = 1  + (−0.546 + 0.837i)2-s + (0.999 + 0.0135i)3-s + (−0.403 − 0.914i)4-s + (0.943 − 0.331i)5-s + (−0.557 + 0.830i)6-s + (−0.860 + 0.508i)7-s + (0.986 + 0.161i)8-s + (0.999 + 0.0270i)9-s + (−0.237 + 0.971i)10-s + (−0.340 − 0.940i)11-s + (−0.391 − 0.920i)12-s + (0.648 − 0.760i)13-s + (0.0439 − 0.999i)14-s + (0.947 − 0.318i)15-s + (−0.674 + 0.738i)16-s + (−0.921 − 0.388i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(961\)    =    \(31^{2}\)
Sign: $0.996 + 0.0857i$
Analytic conductor: \(4.46286\)
Root analytic conductor: \(4.46286\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{961} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 961,\ (0:\ ),\ 0.996 + 0.0857i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.686400372 + 0.07244845121i\)
\(L(\frac12)\) \(\approx\) \(1.686400372 + 0.07244845121i\)
\(L(1)\) \(\approx\) \(1.191134094 + 0.2138065513i\)
\(L(1)\) \(\approx\) \(1.191134094 + 0.2138065513i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad31 \( 1 \)
good2 \( 1 + (-0.546 + 0.837i)T \)
3 \( 1 + (0.999 + 0.0135i)T \)
5 \( 1 + (0.943 - 0.331i)T \)
7 \( 1 + (-0.860 + 0.508i)T \)
11 \( 1 + (-0.340 - 0.940i)T \)
13 \( 1 + (0.648 - 0.760i)T \)
17 \( 1 + (-0.921 - 0.388i)T \)
19 \( 1 + (-0.968 + 0.247i)T \)
23 \( 1 + (0.385 + 0.922i)T \)
29 \( 1 + (0.996 - 0.0809i)T \)
37 \( 1 + (0.283 - 0.959i)T \)
41 \( 1 + (0.540 + 0.841i)T \)
43 \( 1 + (0.850 - 0.526i)T \)
47 \( 1 + (0.230 - 0.972i)T \)
53 \( 1 + (0.907 + 0.419i)T \)
59 \( 1 + (0.334 + 0.942i)T \)
61 \( 1 + (-0.874 - 0.485i)T \)
67 \( 1 + (-0.905 + 0.425i)T \)
71 \( 1 + (0.929 - 0.369i)T \)
73 \( 1 + (0.446 - 0.894i)T \)
79 \( 1 + (0.446 + 0.894i)T \)
83 \( 1 + (-0.452 - 0.891i)T \)
89 \( 1 + (-0.579 - 0.814i)T \)
97 \( 1 + (-0.211 - 0.977i)T \)
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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

−21.42580449483946361171664953149, −20.88196772174895463484389334483, −20.188693163136482849939713349030, −19.39677640740767155641798335451, −18.81696152696841674395429203071, −17.99433396917594451027051780261, −17.276495297647433652662253420456, −16.352066861223766401625747989438, −15.41653595503096924211276376769, −14.34036382862131097516772383703, −13.57312686615836333223651505336, −13.00434901915634338651479000101, −12.41745545610768301560386856867, −10.809085578046994039337087400037, −10.43132393388078600639436869171, −9.516049243658656571180568914762, −9.04212688136588810689799355832, −8.1415461411955047901208495543, −6.912472293635168719224877736668, −6.53671622530605355076405918493, −4.57813667453514325849073060185, −3.97501417197934428618234546041, −2.736867512543513995245109453646, −2.29815377447739864929526980045, −1.2541264851193506480876026502, 0.84774561051136818362781854721, 2.11937657607285047684359101450, 2.99568100264954890918421651928, 4.26551306290182403768347784129, 5.50938498381363409654583292370, 6.11048541807251804823588338275, 6.97725063972679029309863553535, 8.12858479586708399924811574088, 8.82694051229009395021598725762, 9.2326659788605056159128695076, 10.17899288283682034288842079787, 10.84043553361727159474721209915, 12.607923722376543660989374667217, 13.4727858186484640471280216943, 13.620053993910531601539415097950, 14.77995666069554213813829043294, 15.592740649164348636073645581101, 16.068770281962247376507736274822, 16.95959326488886086572669602527, 18.04689406848792664421829784721, 18.4342094217216442322945694046, 19.46226009214545422594844335050, 19.86105242934063591978386783630, 21.07577578025786264812581713855, 21.6024509341025346455563768210

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