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Properties

Label 2-277-277.117-c1-0-10
Degree $2$
Conductor $277$
Sign $0.928 + 0.370i$
Analytic cond. $2.21185$
Root an. cond. $1.48723$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.773i·2-s + (−1.60 − 2.78i)3-s + 1.40·4-s + (3.15 + 1.81i)5-s + (2.15 − 1.24i)6-s + (0.838 − 1.45i)7-s + 2.63i·8-s + (−3.66 + 6.35i)9-s + (−1.40 + 2.43i)10-s + (−2.79 − 1.61i)11-s + (−2.25 − 3.90i)12-s + 6.38·13-s + (1.12 + 0.648i)14-s − 11.7i·15-s + 0.765·16-s + (−3.57 − 2.06i)17-s + ⋯
L(s)  = 1  + 0.547i·2-s + (−0.928 − 1.60i)3-s + 0.700·4-s + (1.40 + 0.813i)5-s + (0.879 − 0.507i)6-s + (0.316 − 0.548i)7-s + 0.930i·8-s + (−1.22 + 2.11i)9-s + (−0.445 + 0.771i)10-s + (−0.843 − 0.486i)11-s + (−0.650 − 1.12i)12-s + 1.77·13-s + (0.300 + 0.173i)14-s − 3.02i·15-s + 0.191·16-s + (−0.867 − 0.500i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 277 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.928 + 0.370i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 277 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.928 + 0.370i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(277\)
Sign: $0.928 + 0.370i$
Analytic conductor: \(2.21185\)
Root analytic conductor: \(1.48723\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{277} (117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 277,\ (\ :1/2),\ 0.928 + 0.370i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39677 - 0.268301i\)
\(L(\frac12)\) \(\approx\) \(1.39677 - 0.268301i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad277 \( 1 + (-7.29 + 14.9i)T \)
good2 \( 1 - 0.773iT - 2T^{2} \)
3 \( 1 + (1.60 + 2.78i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-3.15 - 1.81i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.838 + 1.45i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.79 + 1.61i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 6.38T + 13T^{2} \)
17 \( 1 + (3.57 + 2.06i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 - 0.991T + 19T^{2} \)
23 \( 1 + (-0.177 + 0.307i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.50 + 7.80i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.13 - 1.23i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 0.542iT - 37T^{2} \)
41 \( 1 + 7.78T + 41T^{2} \)
43 \( 1 + (3.61 + 2.08i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.83 - 3.18i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.54 - 0.891i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 3.56T + 59T^{2} \)
61 \( 1 - 7.73iT - 61T^{2} \)
67 \( 1 + (0.509 - 0.881i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (7.35 - 12.7i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 3.54iT - 73T^{2} \)
79 \( 1 + (1.57 + 2.73i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.75 - 13.4i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (7.81 + 13.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.815 - 0.470i)T + (48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.54869805150564278925656900622, −11.05492315115160770000887396917, −10.41870074307199299045747564293, −8.510889752710246318790207330484, −7.48410281495679865503414757682, −6.74551133631989609282009192428, −6.01792199011458798485176766761, −5.52338356094699826993378200449, −2.61214796839267322661763221843, −1.54734054109487739192858047265, 1.76466616170454542669401028176, 3.51034648155860381086622602356, 4.92374663789241904039695782214, 5.67346294391816238256811869584, 6.43193131018128599181730890473, 8.676492359449129130745745496684, 9.359173848229251714592070427643, 10.34566426013731663477830520744, 10.78295403947913847952648309304, 11.62014551640792196953576068492

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