↑

Properties

Label 2-76-4.3-c4-0-27
Degree $2$
Conductor $76$
Sign $0.669 + 0.742i$
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  + (3.73 − 1.43i)2-s − 1.81i·3-s + (11.8 − 10.7i)4-s + 19.2·5-s + (−2.60 − 6.78i)6-s + 41.3i·7-s + (28.9 − 57.0i)8-s + 77.6·9-s + (71.8 − 27.5i)10-s − 163. i·11-s + (−19.4 − 21.6i)12-s − 241.·13-s + (59.3 + 154. i)14-s − 34.9i·15-s + (26.3 − 254. i)16-s + 315.·17-s + ⋯
L(s)  = 1  + (0.933 − 0.358i)2-s − 0.202i·3-s + (0.742 − 0.669i)4-s + 0.769·5-s + (−0.0724 − 0.188i)6-s + 0.844i·7-s + (0.453 − 0.891i)8-s + 0.959·9-s + (0.718 − 0.275i)10-s − 1.35i·11-s + (−0.135 − 0.150i)12-s − 1.42·13-s + (0.302 + 0.788i)14-s − 0.155i·15-s + (0.103 − 0.994i)16-s + 1.09·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.669 + 0.742i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.669 + 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.669 + 0.742i$
Analytic conductor: \(7.85611\)
Root analytic conductor: \(2.80287\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :2),\ 0.669 + 0.742i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.93437 - 1.30517i\)
\(L(\frac12)\) \(\approx\) \(2.93437 - 1.30517i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-3.73 + 1.43i)T \)
19 \( 1 - 82.8iT \)
good3 \( 1 + 1.81iT - 81T^{2} \)
5 \( 1 - 19.2T + 625T^{2} \)
7 \( 1 - 41.3iT - 2.40e3T^{2} \)
11 \( 1 + 163. iT - 1.46e4T^{2} \)
13 \( 1 + 241.T + 2.85e4T^{2} \)
17 \( 1 - 315.T + 8.35e4T^{2} \)
23 \( 1 - 743. iT - 2.79e5T^{2} \)
29 \( 1 + 120.T + 7.07e5T^{2} \)
31 \( 1 - 1.05e3iT - 9.23e5T^{2} \)
37 \( 1 + 2.44e3T + 1.87e6T^{2} \)
41 \( 1 - 1.16e3T + 2.82e6T^{2} \)
43 \( 1 - 3.40e3iT - 3.41e6T^{2} \)
47 \( 1 + 1.30e3iT - 4.87e6T^{2} \)
53 \( 1 + 4.03e3T + 7.89e6T^{2} \)
59 \( 1 - 4.68e3iT - 1.21e7T^{2} \)
61 \( 1 - 1.68e3T + 1.38e7T^{2} \)
67 \( 1 + 3.38e3iT - 2.01e7T^{2} \)
71 \( 1 + 6.80e3iT - 2.54e7T^{2} \)
73 \( 1 + 3.35e3T + 2.83e7T^{2} \)
79 \( 1 + 986. iT - 3.89e7T^{2} \)
83 \( 1 + 8.05e3iT - 4.74e7T^{2} \)
89 \( 1 - 7.97e3T + 6.27e7T^{2} \)
97 \( 1 + 2.36e3T + 8.85e7T^{2} \)
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

−13.60865449298816036011206844558, −12.54894758913542294163974792494, −11.78305030971526895995815356396, −10.33649111944985986540509626604, −9.426651596902267134721024462776, −7.51074317266480622855131460047, −6.04246290175838279934563377877, −5.14669220053782926729082125678, −3.19563386049500878853423435293, −1.64366332527115773802140949496, 2.11942988042214251184381539540, 4.14185851356585704618420830945, 5.16597852812755208041685201687, 6.85584253222911437359741616442, 7.57707833350336117702880680444, 9.746279439017412601970219548021, 10.42374590061705499095157689537, 12.21458509751756052904704972218, 12.83282892172373001635535853214, 14.03621751820660083611094669115

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