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Properties

Label 1-277-277.154-r0-0-0
Degree $1$
Conductor $277$
Sign $-0.967 + 0.251i$
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  + (−0.0682 − 0.997i)2-s + (0.746 − 0.665i)3-s + (−0.990 + 0.136i)4-s + (−0.648 − 0.761i)5-s + (−0.715 − 0.699i)6-s + (0.983 + 0.181i)7-s + (0.203 + 0.979i)8-s + (0.113 − 0.993i)9-s + (−0.715 + 0.699i)10-s + (−0.974 + 0.225i)11-s + (−0.648 + 0.761i)12-s + (−0.917 − 0.398i)13-s + (0.113 − 0.993i)14-s + (−0.990 − 0.136i)15-s + (0.962 − 0.269i)16-s + (−0.419 − 0.907i)17-s + ⋯
L(s)  = 1  + (−0.0682 − 0.997i)2-s + (0.746 − 0.665i)3-s + (−0.990 + 0.136i)4-s + (−0.648 − 0.761i)5-s + (−0.715 − 0.699i)6-s + (0.983 + 0.181i)7-s + (0.203 + 0.979i)8-s + (0.113 − 0.993i)9-s + (−0.715 + 0.699i)10-s + (−0.974 + 0.225i)11-s + (−0.648 + 0.761i)12-s + (−0.917 − 0.398i)13-s + (0.113 − 0.993i)14-s + (−0.990 − 0.136i)15-s + (0.962 − 0.269i)16-s + (−0.419 − 0.907i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 277 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.967 + 0.251i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 277 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.967 + 0.251i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(277\)
Sign: $-0.967 + 0.251i$
Analytic conductor: \(1.28638\)
Root analytic conductor: \(1.28638\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{277} (154, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 277,\ (0:\ ),\ -0.967 + 0.251i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1277406588 - 1.000448213i\)
\(L(\frac12)\) \(\approx\) \(-0.1277406588 - 1.000448213i\)
\(L(1)\) \(\approx\) \(0.5644112148 - 0.8064755357i\)
\(L(1)\) \(\approx\) \(0.5644112148 - 0.8064755357i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad277 \( 1 \)
good2 \( 1 + (-0.0682 - 0.997i)T \)
3 \( 1 + (0.746 - 0.665i)T \)
5 \( 1 + (-0.648 - 0.761i)T \)
7 \( 1 + (0.983 + 0.181i)T \)
11 \( 1 + (-0.974 + 0.225i)T \)
13 \( 1 + (-0.917 - 0.398i)T \)
17 \( 1 + (-0.419 - 0.907i)T \)
19 \( 1 + (-0.334 - 0.942i)T \)
23 \( 1 + (-0.648 - 0.761i)T \)
29 \( 1 + (0.377 + 0.926i)T \)
31 \( 1 + (0.983 - 0.181i)T \)
37 \( 1 + (-0.990 + 0.136i)T \)
41 \( 1 + (-0.576 + 0.816i)T \)
43 \( 1 + (0.898 + 0.439i)T \)
47 \( 1 + (0.995 + 0.0909i)T \)
53 \( 1 + (-0.158 - 0.987i)T \)
59 \( 1 + (0.682 - 0.730i)T \)
61 \( 1 + (0.682 - 0.730i)T \)
67 \( 1 + (-0.247 + 0.968i)T \)
71 \( 1 + (-0.949 - 0.313i)T \)
73 \( 1 + (0.962 + 0.269i)T \)
79 \( 1 + (0.983 - 0.181i)T \)
83 \( 1 + (-0.715 - 0.699i)T \)
89 \( 1 + (-0.998 + 0.0455i)T \)
97 \( 1 + (0.746 - 0.665i)T \)
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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

−26.31783236208910706539244109183, −25.412158058352047254607985506675, −24.26982235474149607873343573584, −23.70072164602830976116318026264, −22.58472814717822267462758109734, −21.68501779776085316130524825418, −20.89011543590929620500031181583, −19.47102799072547264983542186986, −18.92368730820124719074944697613, −17.761475204820436900980912146181, −16.835759989833310253332885174318, −15.56025355713226926945130353517, −15.270271798705121042744770871975, −14.24812006281359803960384892930, −13.73555891538084575349259225004, −12.14626495750345727595202788683, −10.651228743208889428295154328, −10.088691660554017937478884408193, −8.604496534549067005190251179300, −7.9440360197733658497056295072, −7.25600630940545822043811882022, −5.70383924201418512807152212266, −4.51905826737799322194306304236, −3.78953777303985821329421394459, −2.24794574463057994449021392679, 0.63368046221226621806946689928, 2.07240486766637178257390977427, 2.906722408618024454115981760546, 4.46542078232453491080269381991, 5.10784554986927336374565753562, 7.27935992188468270920979573654, 8.18866535512589770851960442264, 8.7557051743251108747941637186, 9.938771236115171773780214835897, 11.24184389102097174372117131775, 12.15309511219983257850813366533, 12.790170034936018889844175230113, 13.74412910871598189313754380863, 14.72979680369756353783042812010, 15.73472154356505223478278339270, 17.39163386922200971619691329728, 18.00507274052309680662257108056, 18.95987978621956159553754671979, 19.86616151147039183228722767498, 20.49705243137371638841493221187, 21.081109086521313142110595299517, 22.32956757500355908567470701337, 23.600096159105744029926042358565, 24.09056071002577191447002101774, 25.02641404039885894527765189968

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