↑

Properties

Label 2-76-19.8-c4-0-1
Degree $2$
Conductor $76$
Sign $0.996 + 0.0876i$
Analytic cond. $7.85611$
Root an. cond. $2.80287$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−9.05 − 5.22i)3-s + (−12.7 + 22.0i)5-s + 27.7·7-s + (14.1 + 24.4i)9-s + 143.·11-s + (123. − 71.5i)13-s + (230. − 133. i)15-s + (43.9 − 76.1i)17-s + (340. − 119. i)19-s + (−251. − 145. i)21-s + (350. + 607. i)23-s + (−12.6 − 21.9i)25-s + 551. i·27-s + (−37.5 + 21.6i)29-s + 111. i·31-s + ⋯
L(s)  = 1  + (−1.00 − 0.580i)3-s + (−0.510 + 0.883i)5-s + 0.566·7-s + (0.174 + 0.301i)9-s + 1.18·11-s + (0.733 − 0.423i)13-s + (1.02 − 0.592i)15-s + (0.152 − 0.263i)17-s + (0.943 − 0.331i)19-s + (−0.569 − 0.328i)21-s + (0.663 + 1.14i)23-s + (−0.0202 − 0.0350i)25-s + 0.756i·27-s + (−0.0446 + 0.0257i)29-s + 0.115i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0876i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.996 + 0.0876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.996 + 0.0876i$
Analytic conductor: \(7.85611\)
Root analytic conductor: \(2.80287\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :2),\ 0.996 + 0.0876i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.20619 - 0.0529882i\)
\(L(\frac12)\) \(\approx\) \(1.20619 - 0.0529882i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + (-340. + 119. i)T \)
good3 \( 1 + (9.05 + 5.22i)T + (40.5 + 70.1i)T^{2} \)
5 \( 1 + (12.7 - 22.0i)T + (-312.5 - 541. i)T^{2} \)
7 \( 1 - 27.7T + 2.40e3T^{2} \)
11 \( 1 - 143.T + 1.46e4T^{2} \)
13 \( 1 + (-123. + 71.5i)T + (1.42e4 - 2.47e4i)T^{2} \)
17 \( 1 + (-43.9 + 76.1i)T + (-4.17e4 - 7.23e4i)T^{2} \)
23 \( 1 + (-350. - 607. i)T + (-1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + (37.5 - 21.6i)T + (3.53e5 - 6.12e5i)T^{2} \)
31 \( 1 - 111. iT - 9.23e5T^{2} \)
37 \( 1 + 1.07e3iT - 1.87e6T^{2} \)
41 \( 1 + (-80.1 - 46.2i)T + (1.41e6 + 2.44e6i)T^{2} \)
43 \( 1 + (-945. + 1.63e3i)T + (-1.70e6 - 2.96e6i)T^{2} \)
47 \( 1 + (-1.36e3 - 2.36e3i)T + (-2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 + (-2.15e3 + 1.24e3i)T + (3.94e6 - 6.83e6i)T^{2} \)
59 \( 1 + (5.03e3 + 2.90e3i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (-1.26e3 - 2.18e3i)T + (-6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (-4.30e3 + 2.48e3i)T + (1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 + (-3.72e3 - 2.14e3i)T + (1.27e7 + 2.20e7i)T^{2} \)
73 \( 1 + (1.58e3 - 2.75e3i)T + (-1.41e7 - 2.45e7i)T^{2} \)
79 \( 1 + (-4.93e3 - 2.84e3i)T + (1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 - 1.04e4T + 4.74e7T^{2} \)
89 \( 1 + (-4.82e3 + 2.78e3i)T + (3.13e7 - 5.43e7i)T^{2} \)
97 \( 1 + (1.56e4 + 9.01e3i)T + (4.42e7 + 7.66e7i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.77994798541360127008658526056, −12.36346592582122314543504251610, −11.39012054044051310304670837252, −11.00737805305352621968111736633, −9.234042062099957603372814341828, −7.57001237676176427288388465941, −6.67887692065854510134138406173, −5.43930753456974575782447389636, −3.51987128787946153539247403895, −1.08632007970263227989698715928, 1.02476911558368316597345904813, 4.08141554617367020799822239396, 5.04746703092719019622758972909, 6.39700248830826099957006643414, 8.161241036507526967929948745613, 9.245995624130033492946102757797, 10.69489645835346391953521520891, 11.62959498548150221315716898935, 12.28085872083157927092952683245, 13.80778387204031005806114998640

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