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Properties

Label 2-127-127.107-c1-0-8
Degree $2$
Conductor $127$
Sign $-0.753 + 0.657i$
Analytic cond. $1.01410$
Root an. cond. $1.00702$
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  + 0.534·2-s + (−1.38 − 2.40i)3-s − 1.71·4-s − 0.988·5-s + (−0.741 − 1.28i)6-s + (−0.163 − 0.283i)7-s − 1.98·8-s + (−2.34 + 4.06i)9-s − 0.528·10-s + (2.49 − 4.31i)11-s + (2.37 + 4.11i)12-s + (−1.76 − 3.05i)13-s + (−0.0876 − 0.151i)14-s + (1.37 + 2.37i)15-s + 2.36·16-s + (0.276 − 0.479i)17-s + ⋯
L(s)  = 1  + 0.378·2-s + (−0.800 − 1.38i)3-s − 0.857·4-s − 0.442·5-s + (−0.302 − 0.524i)6-s + (−0.0619 − 0.107i)7-s − 0.702·8-s + (−0.781 + 1.35i)9-s − 0.167·10-s + (0.751 − 1.30i)11-s + (0.686 + 1.18i)12-s + (−0.489 − 0.847i)13-s + (−0.0234 − 0.0405i)14-s + (0.353 + 0.612i)15-s + 0.591·16-s + (0.0671 − 0.116i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.753 + 0.657i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.753 + 0.657i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(127\)
Sign: $-0.753 + 0.657i$
Analytic conductor: \(1.01410\)
Root analytic conductor: \(1.00702\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{127} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 127,\ (\ :1/2),\ -0.753 + 0.657i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.227731 - 0.606954i\)
\(L(\frac12)\) \(\approx\) \(0.227731 - 0.606954i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad127 \( 1 + (-11.1 - 1.90i)T \)
good2 \( 1 - 0.534T + 2T^{2} \)
3 \( 1 + (1.38 + 2.40i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 0.988T + 5T^{2} \)
7 \( 1 + (0.163 + 0.283i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.49 + 4.31i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.76 + 3.05i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.276 + 0.479i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 - 5.21T + 19T^{2} \)
23 \( 1 + (-2.05 - 3.55i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.93 + 5.08i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.61 + 4.52i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.37 - 7.57i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.970 - 1.68i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.43 - 4.20i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 2.39T + 47T^{2} \)
53 \( 1 + (-1.39 + 2.42i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.81 + 11.8i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 - 2.72T + 61T^{2} \)
67 \( 1 + (-4.67 - 8.10i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-5.95 - 10.3i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 14.1T + 73T^{2} \)
79 \( 1 + (7.78 + 13.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8.60 + 14.9i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 16.4T + 89T^{2} \)
97 \( 1 + (-0.114 - 0.197i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.18523517953485325896286651443, −11.82146957916539060855051243888, −11.61259935238625341745768238048, −9.853530440350216320377511633514, −8.406091123394962885499039906123, −7.48608237652521181929628229055, −6.13742954279660653295469980484, −5.26882873108005251697064355407, −3.44957468806747411070369550387, −0.71690980059883717742189138441, 3.75630639326906263350688761614, 4.57396528111464169400528195903, 5.44371349817367074576975325889, 7.06022392683605950246992832972, 8.973607063014128277508104761760, 9.587874566515964131967163340256, 10.56654080943343223449120320844, 11.93191639123894890650216544622, 12.33096485844965908668893546833, 14.03411059728638742103802820479

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