Properties

Label 2-76-76.7-c2-0-4
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: $\chi_{76} (7, \cdot )$
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

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

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