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Properties

Label 1-277-277.155-r0-0-0
Degree $1$
Conductor $277$
Sign $0.446 - 0.894i$
Analytic cond. $1.28638$
Root an. cond. $1.28638$
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  + (−0.917 − 0.398i)2-s + (−0.334 − 0.942i)3-s + (0.682 + 0.730i)4-s + (0.460 + 0.887i)5-s + (−0.0682 + 0.997i)6-s + (0.460 − 0.887i)7-s + (−0.334 − 0.942i)8-s + (−0.775 + 0.631i)9-s + (−0.0682 − 0.997i)10-s + (0.203 + 0.979i)11-s + (0.460 − 0.887i)12-s + (−0.775 − 0.631i)13-s + (−0.775 + 0.631i)14-s + (0.682 − 0.730i)15-s + (−0.0682 + 0.997i)16-s + (0.854 − 0.519i)17-s + ⋯
L(s)  = 1  + (−0.917 − 0.398i)2-s + (−0.334 − 0.942i)3-s + (0.682 + 0.730i)4-s + (0.460 + 0.887i)5-s + (−0.0682 + 0.997i)6-s + (0.460 − 0.887i)7-s + (−0.334 − 0.942i)8-s + (−0.775 + 0.631i)9-s + (−0.0682 − 0.997i)10-s + (0.203 + 0.979i)11-s + (0.460 − 0.887i)12-s + (−0.775 − 0.631i)13-s + (−0.775 + 0.631i)14-s + (0.682 − 0.730i)15-s + (−0.0682 + 0.997i)16-s + (0.854 − 0.519i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 277 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.446 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 277 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.446 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(277\)
Sign: $0.446 - 0.894i$
Analytic conductor: \(1.28638\)
Root analytic conductor: \(1.28638\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{277} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 277,\ (0:\ ),\ 0.446 - 0.894i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6920432829 - 0.4280490153i\)
\(L(\frac12)\) \(\approx\) \(0.6920432829 - 0.4280490153i\)
\(L(1)\) \(\approx\) \(0.6955655867 - 0.2676278838i\)
\(L(1)\) \(\approx\) \(0.6955655867 - 0.2676278838i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad277 \( 1 \)
good2 \( 1 + (-0.917 - 0.398i)T \)
3 \( 1 + (-0.334 - 0.942i)T \)
5 \( 1 + (0.460 + 0.887i)T \)
7 \( 1 + (0.460 - 0.887i)T \)
11 \( 1 + (0.203 + 0.979i)T \)
13 \( 1 + (-0.775 - 0.631i)T \)
17 \( 1 + (0.854 - 0.519i)T \)
19 \( 1 + (0.460 - 0.887i)T \)
23 \( 1 + (0.460 + 0.887i)T \)
29 \( 1 + (0.682 - 0.730i)T \)
31 \( 1 + (0.460 + 0.887i)T \)
37 \( 1 + (0.682 + 0.730i)T \)
41 \( 1 + (0.854 - 0.519i)T \)
43 \( 1 + (-0.917 - 0.398i)T \)
47 \( 1 + (0.854 - 0.519i)T \)
53 \( 1 + (-0.576 - 0.816i)T \)
59 \( 1 + (0.203 - 0.979i)T \)
61 \( 1 + (0.203 - 0.979i)T \)
67 \( 1 + (-0.0682 + 0.997i)T \)
71 \( 1 + (-0.334 - 0.942i)T \)
73 \( 1 + (-0.0682 - 0.997i)T \)
79 \( 1 + (0.460 + 0.887i)T \)
83 \( 1 + (-0.0682 + 0.997i)T \)
89 \( 1 + (0.962 + 0.269i)T \)
97 \( 1 + (-0.334 - 0.942i)T \)
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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

−25.91335391552799893174299213976, −24.88151072538405858628274755937, −24.36770311651182854476095996913, −23.30974683927172086422393533337, −21.849515092393931263582757624374, −21.24743553273798832719663771735, −20.4692860444885236790604833205, −19.31837455333364409761555720637, −18.3527155345166304071934361231, −17.28999226906729021898142917796, −16.569589575030282052844538432734, −16.10296194341439252747369976630, −14.80853921653605120812730634766, −14.24196700019643054483422223796, −12.31392754928407076066455229199, −11.58021209509845861840516057716, −10.437519282477667712969971396497, −9.48518951996015423162230417287, −8.83132386535200834952703446011, −7.99059574203031860893184002341, −6.13284787478505559294277034154, −5.589537848856491566622925979488, −4.5394568937062764715938336125, −2.69224420076506017641406480949, −1.1601228386442135920235527343, 0.97758598148655575409587851775, 2.167546903478971461021046815961, 3.17861369919078452903960580466, 5.0856793557904127501180501195, 6.66538413045745823054863663025, 7.28388772321326880934469415163, 7.92719738996038032057834010739, 9.595949332685420597732312334847, 10.33563775834484246836403548981, 11.3171688862095169255468074314, 12.10224933649245476223971549912, 13.24703366224117308118506102131, 14.19312891346666311341716670719, 15.35621310395444531191099035666, 16.87730038205845021556216186338, 17.641406297388019984868860222433, 17.87267545568939908831506728142, 19.07229634775228599838694650445, 19.7859103974611234049506862895, 20.67043004551563633976579765759, 21.88280004418637774400896775079, 22.79768747962020206820373768707, 23.68183052942550176094715853315, 25.051950956204471002499936601787, 25.339758956008083417819445625803

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