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Properties

Label 1-31e2-961.284-r0-0-0
Degree $1$
Conductor $961$
Sign $-0.396 + 0.917i$
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  + (−0.874 − 0.485i)2-s + (−0.378 + 0.925i)3-s + (0.528 + 0.848i)4-s + (0.283 + 0.959i)5-s + (0.780 − 0.625i)6-s + (0.409 + 0.912i)7-s + (−0.0506 − 0.998i)8-s + (−0.713 − 0.701i)9-s + (0.217 − 0.975i)10-s + (−0.184 − 0.982i)11-s + (−0.985 + 0.168i)12-s + (0.963 + 0.266i)13-s + (0.0843 − 0.996i)14-s + (−0.994 − 0.101i)15-s + (−0.440 + 0.897i)16-s + (−0.315 − 0.948i)17-s + ⋯
L(s)  = 1  + (−0.874 − 0.485i)2-s + (−0.378 + 0.925i)3-s + (0.528 + 0.848i)4-s + (0.283 + 0.959i)5-s + (0.780 − 0.625i)6-s + (0.409 + 0.912i)7-s + (−0.0506 − 0.998i)8-s + (−0.713 − 0.701i)9-s + (0.217 − 0.975i)10-s + (−0.184 − 0.982i)11-s + (−0.985 + 0.168i)12-s + (0.963 + 0.266i)13-s + (0.0843 − 0.996i)14-s + (−0.994 − 0.101i)15-s + (−0.440 + 0.897i)16-s + (−0.315 − 0.948i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.396 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.396 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(961\)    =    \(31^{2}\)
Sign: $-0.396 + 0.917i$
Analytic conductor: \(4.46286\)
Root analytic conductor: \(4.46286\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{961} (284, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 961,\ (0:\ ),\ -0.396 + 0.917i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4668819256 + 0.7105151256i\)
\(L(\frac12)\) \(\approx\) \(0.4668819256 + 0.7105151256i\)
\(L(1)\) \(\approx\) \(0.6530111554 + 0.2796678687i\)
\(L(1)\) \(\approx\) \(0.6530111554 + 0.2796678687i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad31 \( 1 \)
good2 \( 1 + (-0.874 - 0.485i)T \)
3 \( 1 + (-0.378 + 0.925i)T \)
5 \( 1 + (0.283 + 0.959i)T \)
7 \( 1 + (0.409 + 0.912i)T \)
11 \( 1 + (-0.184 - 0.982i)T \)
13 \( 1 + (0.963 + 0.266i)T \)
17 \( 1 + (-0.315 - 0.948i)T \)
19 \( 1 + (-0.117 - 0.993i)T \)
23 \( 1 + (0.688 + 0.724i)T \)
29 \( 1 + (0.688 + 0.724i)T \)
37 \( 1 + (-0.713 + 0.701i)T \)
41 \( 1 + (0.890 + 0.455i)T \)
43 \( 1 + (0.217 + 0.975i)T \)
47 \( 1 + (0.688 + 0.724i)T \)
53 \( 1 + (0.990 - 0.134i)T \)
59 \( 1 + (-0.713 + 0.701i)T \)
61 \( 1 + (0.347 + 0.937i)T \)
67 \( 1 + (-0.664 + 0.747i)T \)
71 \( 1 + (-0.117 + 0.993i)T \)
73 \( 1 + (-0.905 + 0.425i)T \)
79 \( 1 + (-0.905 - 0.425i)T \)
83 \( 1 + (0.857 - 0.514i)T \)
89 \( 1 + (-0.994 + 0.101i)T \)
97 \( 1 + (0.528 - 0.848i)T \)
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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

−21.10193483658479009969583237839, −20.51414755853215126576066289723, −19.87417083034192376810804155027, −19.04186881471354444588053090956, −18.14359323492806752412630180622, −17.46154176420461029305233350615, −17.07387949674601286133803178972, −16.32582217608920280537728283978, −15.371438175428616648474820290690, −14.29388208336345441056281679099, −13.55519174399417894258590593221, −12.68356152323078148284285077422, −11.94406669719998405919109231825, −10.68755673433670656486848195268, −10.39919716708607440233700298499, −9.05820515742987990006024319224, −8.27741646282338028836602277522, −7.71723516649638764102999927799, −6.78618383504906511949327427219, −5.98551802916179103549763866496, −5.15299797321107155348929164263, −4.11087228848013891777050024174, −2.14080517672150963761263440117, −1.49125121725779282427088202209, −0.59291235299623090618689143120, 1.19736860280490838818109731942, 2.837525451951296876180411034598, 2.96348101565727095187181011806, 4.31328771826032491324802577875, 5.55315456706185325079832374375, 6.33169618437446877144483648500, 7.28806870022242374374025110124, 8.67702822638461880113507224279, 8.9592645292867920994519494129, 9.92967323794107746818281495561, 10.89900852470517221739787103306, 11.260674806633922073161098807360, 11.806808968328122622120964919768, 13.20708643839111443771655217170, 14.13142389877512080472586235099, 15.22653461769725497597804618519, 15.793477811197234520620655475597, 16.43679890388705569922118731687, 17.65327098379711960211572067375, 17.95270342278409144233169089469, 18.78872458471269661094381430642, 19.49185376319398732205332610240, 20.6606439348268543533147023834, 21.32125483523728207014112282179, 21.730672945519923571764022674101

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