Properties

Label 2-76-76.71-c1-0-4
Degree $2$
Conductor $76$
Sign $0.922 - 0.385i$
Analytic cond. $0.606863$
Root an. cond. $0.779014$
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  + (0.927 + 1.06i)2-s + (0.306 − 1.73i)3-s + (−0.280 + 1.98i)4-s + (−0.220 − 0.184i)5-s + (2.14 − 1.28i)6-s + (−0.588 + 0.339i)7-s + (−2.37 + 1.53i)8-s + (−0.110 − 0.0403i)9-s + (−0.00686 − 0.406i)10-s + (−3.85 − 2.22i)11-s + (3.35 + 1.09i)12-s + (−3.41 + 0.602i)13-s + (−0.908 − 0.313i)14-s + (−0.388 + 0.326i)15-s + (−3.84 − 1.11i)16-s + (4.15 − 1.51i)17-s + ⋯
L(s)  = 1  + (0.655 + 0.755i)2-s + (0.177 − 1.00i)3-s + (−0.140 + 0.990i)4-s + (−0.0984 − 0.0826i)5-s + (0.874 − 0.524i)6-s + (−0.222 + 0.128i)7-s + (−0.839 + 0.543i)8-s + (−0.0369 − 0.0134i)9-s + (−0.00217 − 0.128i)10-s + (−1.16 − 0.670i)11-s + (0.969 + 0.316i)12-s + (−0.947 + 0.166i)13-s + (−0.242 − 0.0837i)14-s + (−0.100 + 0.0842i)15-s + (−0.960 − 0.277i)16-s + (1.00 − 0.366i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 - 0.385i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.922 - 0.385i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.922 - 0.385i$
Analytic conductor: \(0.606863\)
Root analytic conductor: \(0.779014\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :1/2),\ 0.922 - 0.385i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19736 + 0.240042i\)
\(L(\frac12)\) \(\approx\) \(1.19736 + 0.240042i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.927 - 1.06i)T \)
19 \( 1 + (-1.76 - 3.98i)T \)
good3 \( 1 + (-0.306 + 1.73i)T + (-2.81 - 1.02i)T^{2} \)
5 \( 1 + (0.220 + 0.184i)T + (0.868 + 4.92i)T^{2} \)
7 \( 1 + (0.588 - 0.339i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (3.85 + 2.22i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.41 - 0.602i)T + (12.2 - 4.44i)T^{2} \)
17 \( 1 + (-4.15 + 1.51i)T + (13.0 - 10.9i)T^{2} \)
23 \( 1 + (-0.347 - 0.413i)T + (-3.99 + 22.6i)T^{2} \)
29 \( 1 + (-1.03 + 2.85i)T + (-22.2 - 18.6i)T^{2} \)
31 \( 1 + (-5.24 - 9.08i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 8.82iT - 37T^{2} \)
41 \( 1 + (-1.85 - 0.326i)T + (38.5 + 14.0i)T^{2} \)
43 \( 1 + (-3.49 + 4.16i)T + (-7.46 - 42.3i)T^{2} \)
47 \( 1 + (-0.419 + 1.15i)T + (-36.0 - 30.2i)T^{2} \)
53 \( 1 + (6.41 + 7.64i)T + (-9.20 + 52.1i)T^{2} \)
59 \( 1 + (4.20 - 1.53i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-6.04 + 5.07i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (4.66 + 1.69i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (6.81 + 5.72i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (-0.591 + 3.35i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (1.43 - 8.15i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (-8.64 + 4.99i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (10.6 - 1.87i)T + (83.6 - 30.4i)T^{2} \)
97 \( 1 + (1.24 + 3.40i)T + (-74.3 + 62.3i)T^{2} \)
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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

−14.28147029982022997107300215430, −13.64691105583682018704918003922, −12.54127639834369960206430509320, −11.98747985971828378191473500290, −10.05868433684998320365688616832, −8.238121818762756193502534691956, −7.59692644328172234266657256422, −6.34526311277669775861497782076, −4.99513542826094963796981883615, −2.90808517350856437628625837204, 2.88054279075999789447174420064, 4.36203839114216384318081832407, 5.43032731156648108350494198832, 7.41495058831046442074467415736, 9.439748296329264472896432670940, 10.05639303412709074110825640299, 11.01070380919014501838126462535, 12.38308413384476384359094263608, 13.21890906858892531670067107832, 14.61002226469882845360837048487

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