Properties

Label 2-76-76.11-c2-0-15
Degree $2$
Conductor $76$
Sign $-0.461 + 0.887i$
Analytic cond. $2.07085$
Root an. cond. $1.43904$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.00657 − 1.99i)2-s + (0.443 + 0.255i)3-s + (−3.99 − 0.0262i)4-s + (1.99 − 3.44i)5-s + (0.514 − 0.884i)6-s − 9.66i·7-s + (−0.0788 + 7.99i)8-s + (−4.36 − 7.56i)9-s + (−6.88 − 4.00i)10-s + 2.62i·11-s + (−1.76 − 1.03i)12-s + (11.8 + 20.4i)13-s + (−19.3 − 0.0635i)14-s + (1.76 − 1.01i)15-s + (15.9 + 0.210i)16-s + (−3.40 + 5.90i)17-s + ⋯
L(s)  = 1  + (0.00328 − 0.999i)2-s + (0.147 + 0.0852i)3-s + (−0.999 − 0.00657i)4-s + (0.398 − 0.689i)5-s + (0.0857 − 0.147i)6-s − 1.38i·7-s + (−0.00985 + 0.999i)8-s + (−0.485 − 0.840i)9-s + (−0.688 − 0.400i)10-s + 0.238i·11-s + (−0.147 − 0.0862i)12-s + (0.907 + 1.57i)13-s + (−1.38 − 0.00453i)14-s + (0.117 − 0.0678i)15-s + (0.999 + 0.0131i)16-s + (−0.200 + 0.347i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.461 + 0.887i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.461 + 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-0.461 + 0.887i$
Analytic conductor: \(2.07085\)
Root analytic conductor: \(1.43904\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :1),\ -0.461 + 0.887i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.632561 - 1.04226i\)
\(L(\frac12)\) \(\approx\) \(0.632561 - 1.04226i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.00657 + 1.99i)T \)
19 \( 1 + (-12.0 + 14.7i)T \)
good3 \( 1 + (-0.443 - 0.255i)T + (4.5 + 7.79i)T^{2} \)
5 \( 1 + (-1.99 + 3.44i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 + 9.66iT - 49T^{2} \)
11 \( 1 - 2.62iT - 121T^{2} \)
13 \( 1 + (-11.8 - 20.4i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + (3.40 - 5.90i)T + (-144.5 - 250. i)T^{2} \)
23 \( 1 + (-17.2 + 9.96i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-0.445 - 0.772i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 - 59.6iT - 961T^{2} \)
37 \( 1 + 7.80T + 1.36e3T^{2} \)
41 \( 1 + (-15.9 + 27.6i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (6.09 + 3.51i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (47.7 - 27.5i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (33.4 + 57.9i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-51.3 - 29.6i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (1.23 + 2.14i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-36.4 + 21.0i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (88.6 + 51.1i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (7.82 - 13.5i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-63.6 - 36.7i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 18.4iT - 6.88e3T^{2} \)
89 \( 1 + (35.4 + 61.4i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (39.9 - 69.2i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−13.74121663147010181277380657885, −12.86969142615539758174218141084, −11.62731375968671870192364839266, −10.67375179119292734937037765232, −9.374747055652684206332659173962, −8.710728018748123892870553617489, −6.77265593503613289547176016622, −4.80902670577060442322202067766, −3.59217343933997716848891317022, −1.22775444372613298787100067357, 2.97247782639393009171419061880, 5.40154448720781128249122022933, 6.06983022324867110219765928032, 7.77330695094379031911358596110, 8.628206071992769341333766757090, 9.921605753574333865923034932284, 11.26678015050797071127434246455, 12.85409816839466541313816089865, 13.71572431913369374352355236802, 14.78833765376522954704929791257

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