Properties

Label 2-76-4.3-c2-0-10
Degree $2$
Conductor $76$
Sign $0.971 + 0.238i$
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  + (1.57 − 1.23i)2-s + 2.90i·3-s + (0.954 − 3.88i)4-s + 3.66·5-s + (3.57 + 4.56i)6-s + 1.93i·7-s + (−3.29 − 7.29i)8-s + 0.583·9-s + (5.76 − 4.51i)10-s + 0.752i·11-s + (11.2 + 2.77i)12-s − 13.0·13-s + (2.38 + 3.04i)14-s + 10.6i·15-s + (−14.1 − 7.41i)16-s − 23.9·17-s + ⋯
L(s)  = 1  + (0.786 − 0.616i)2-s + 0.967i·3-s + (0.238 − 0.971i)4-s + 0.732·5-s + (0.596 + 0.761i)6-s + 0.276i·7-s + (−0.411 − 0.911i)8-s + 0.0647·9-s + (0.576 − 0.451i)10-s + 0.0684i·11-s + (0.939 + 0.230i)12-s − 1.00·13-s + (0.170 + 0.217i)14-s + 0.708i·15-s + (−0.886 − 0.463i)16-s − 1.40·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 + 0.238i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.971 + 0.238i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.971 + 0.238i$
Analytic conductor: \(2.07085\)
Root analytic conductor: \(1.43904\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :1),\ 0.971 + 0.238i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.93215 - 0.233988i\)
\(L(\frac12)\) \(\approx\) \(1.93215 - 0.233988i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.57 + 1.23i)T \)
19 \( 1 + 4.35iT \)
good3 \( 1 - 2.90iT - 9T^{2} \)
5 \( 1 - 3.66T + 25T^{2} \)
7 \( 1 - 1.93iT - 49T^{2} \)
11 \( 1 - 0.752iT - 121T^{2} \)
13 \( 1 + 13.0T + 169T^{2} \)
17 \( 1 + 23.9T + 289T^{2} \)
23 \( 1 + 6.26iT - 529T^{2} \)
29 \( 1 - 33.1T + 841T^{2} \)
31 \( 1 - 17.5iT - 961T^{2} \)
37 \( 1 - 41.5T + 1.36e3T^{2} \)
41 \( 1 + 5.51T + 1.68e3T^{2} \)
43 \( 1 + 84.5iT - 1.84e3T^{2} \)
47 \( 1 - 18.3iT - 2.20e3T^{2} \)
53 \( 1 + 41.2T + 2.80e3T^{2} \)
59 \( 1 - 69.5iT - 3.48e3T^{2} \)
61 \( 1 - 87.6T + 3.72e3T^{2} \)
67 \( 1 + 105. iT - 4.48e3T^{2} \)
71 \( 1 - 74.9iT - 5.04e3T^{2} \)
73 \( 1 - 48.7T + 5.32e3T^{2} \)
79 \( 1 - 95.9iT - 6.24e3T^{2} \)
83 \( 1 + 65.8iT - 6.88e3T^{2} \)
89 \( 1 - 3.38T + 7.92e3T^{2} \)
97 \( 1 + 23.5T + 9.40e3T^{2} \)
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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

−14.17933364613514416580059856527, −13.17392941947383335790069479762, −12.08159945556945188868860057679, −10.79658463765525093292672311159, −9.967421620720150383910022073919, −9.098852490562446322665058907345, −6.73787264743729717906196191461, −5.27969707186342805859828050863, −4.25955615568781187546763519380, −2.41380847537043515686172382039, 2.32594825374206141238343319153, 4.54165649386891724765475278360, 6.11665954184456661816029402931, 7.01846793213364172465418618910, 8.106652815243847404684280132071, 9.717454600907523671515010073641, 11.41252053413062078858783123758, 12.57991034686504322267663032051, 13.30879426630525544152173088834, 14.04090795878538011070980150961

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