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Properties

Label 2-127-127.107-c1-0-5
Degree $2$
Conductor $127$
Sign $0.743 - 0.668i$
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  + 1.15·2-s + (1.20 + 2.08i)3-s − 0.660·4-s + 0.956·5-s + (1.39 + 2.41i)6-s + (−1.35 − 2.35i)7-s − 3.07·8-s + (−1.39 + 2.41i)9-s + 1.10·10-s + (0.927 − 1.60i)11-s + (−0.794 − 1.37i)12-s + (1.21 + 2.11i)13-s + (−1.57 − 2.72i)14-s + (1.14 + 1.99i)15-s − 2.24·16-s + (1.89 − 3.28i)17-s + ⋯
L(s)  = 1  + 0.818·2-s + (0.694 + 1.20i)3-s − 0.330·4-s + 0.427·5-s + (0.568 + 0.983i)6-s + (−0.513 − 0.889i)7-s − 1.08·8-s + (−0.463 + 0.803i)9-s + 0.349·10-s + (0.279 − 0.484i)11-s + (−0.229 − 0.397i)12-s + (0.338 + 0.586i)13-s + (−0.420 − 0.727i)14-s + (0.296 + 0.514i)15-s − 0.560·16-s + (0.459 − 0.795i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.743 - 0.668i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.743 - 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(127\)
Sign: $0.743 - 0.668i$
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: \((2,\ 127,\ (\ :1/2),\ 0.743 - 0.668i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.54198 + 0.590935i\)
\(L(\frac12)\) \(\approx\) \(1.54198 + 0.590935i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad127 \( 1 + (11.1 + 1.74i)T \)
good2 \( 1 - 1.15T + 2T^{2} \)
3 \( 1 + (-1.20 - 2.08i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 0.956T + 5T^{2} \)
7 \( 1 + (1.35 + 2.35i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.927 + 1.60i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.21 - 2.11i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.89 + 3.28i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + 0.695T + 19T^{2} \)
23 \( 1 + (0.861 + 1.49i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.55 - 6.16i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.312 + 0.541i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.42 - 5.92i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.37 - 7.57i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.66 + 8.07i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 8.35T + 47T^{2} \)
53 \( 1 + (4.68 - 8.11i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.97 + 10.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 - 10.2T + 61T^{2} \)
67 \( 1 + (-6.30 - 10.9i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.37 + 9.30i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 10.4T + 73T^{2} \)
79 \( 1 + (-1.71 - 2.97i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.76 - 11.7i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 2.66T + 89T^{2} \)
97 \( 1 + (6.47 + 11.2i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.92028312361377970272590657124, −12.82512011492121639739518226295, −11.42178996401390788245767860629, −10.08337097001334411304410827712, −9.469998644888539319630894823111, −8.498921059306594222111442429218, −6.64974951713132320356283863542, −5.21491224213858945752549691570, −4.03682818832008690326221509162, −3.27323886803086268762925620073, 2.19410045829833011474007397682, 3.63416609184221645235831423934, 5.54966152890971633640815695061, 6.38848074274947339844451186639, 7.87791667003482075848957406947, 8.881019555444675104341190546607, 9.878103331774803576009843675707, 11.80132712649318441289730977476, 12.74368030722091153914733965139, 13.06633316539699179027817186139

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