Properties

Label 2-76-19.14-c4-0-1
Degree $2$
Conductor $76$
Sign $0.661 - 0.749i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−17.2 − 3.03i)3-s + (−15.3 − 12.9i)5-s + (−26.4 − 45.7i)7-s + (211. + 76.8i)9-s + (−106. + 184. i)11-s + (67.2 − 11.8i)13-s + (225. + 269. i)15-s + (86.5 − 31.4i)17-s + (360. − 10.5i)19-s + (316. + 868. i)21-s + (152. − 128. i)23-s + (−38.3 − 217. i)25-s + (−2.17e3 − 1.25e3i)27-s + (−125. + 345. i)29-s + (−715. + 412. i)31-s + ⋯
L(s)  = 1  + (−1.91 − 0.337i)3-s + (−0.615 − 0.516i)5-s + (−0.539 − 0.934i)7-s + (2.60 + 0.948i)9-s + (−0.880 + 1.52i)11-s + (0.398 − 0.0702i)13-s + (1.00 + 1.19i)15-s + (0.299 − 0.108i)17-s + (0.999 − 0.0293i)19-s + (0.716 + 1.96i)21-s + (0.288 − 0.242i)23-s + (−0.0614 − 0.348i)25-s + (−2.98 − 1.72i)27-s + (−0.149 + 0.411i)29-s + (−0.744 + 0.429i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.661 - 0.749i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.661 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.661 - 0.749i$
Analytic conductor: \(7.85611\)
Root analytic conductor: \(2.80287\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :2),\ 0.661 - 0.749i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.396345 + 0.178878i\)
\(L(\frac12)\) \(\approx\) \(0.396345 + 0.178878i\)
\(L(3)\) 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 \( 1 + (-360. + 10.5i)T \)
good3 \( 1 + (17.2 + 3.03i)T + (76.1 + 27.7i)T^{2} \)
5 \( 1 + (15.3 + 12.9i)T + (108. + 615. i)T^{2} \)
7 \( 1 + (26.4 + 45.7i)T + (-1.20e3 + 2.07e3i)T^{2} \)
11 \( 1 + (106. - 184. i)T + (-7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 + (-67.2 + 11.8i)T + (2.68e4 - 9.76e3i)T^{2} \)
17 \( 1 + (-86.5 + 31.4i)T + (6.39e4 - 5.36e4i)T^{2} \)
23 \( 1 + (-152. + 128. i)T + (4.85e4 - 2.75e5i)T^{2} \)
29 \( 1 + (125. - 345. i)T + (-5.41e5 - 4.54e5i)T^{2} \)
31 \( 1 + (715. - 412. i)T + (4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 - 2.46e3iT - 1.87e6T^{2} \)
41 \( 1 + (-1.69e3 - 299. i)T + (2.65e6 + 9.66e5i)T^{2} \)
43 \( 1 + (1.34e3 + 1.12e3i)T + (5.93e5 + 3.36e6i)T^{2} \)
47 \( 1 + (-1.84e3 - 670. i)T + (3.73e6 + 3.13e6i)T^{2} \)
53 \( 1 + (-1.28e3 - 1.52e3i)T + (-1.37e6 + 7.77e6i)T^{2} \)
59 \( 1 + (-836. - 2.29e3i)T + (-9.28e6 + 7.78e6i)T^{2} \)
61 \( 1 + (1.06e3 - 891. i)T + (2.40e6 - 1.36e7i)T^{2} \)
67 \( 1 + (-415. + 1.14e3i)T + (-1.54e7 - 1.29e7i)T^{2} \)
71 \( 1 + (5.93e3 - 7.07e3i)T + (-4.41e6 - 2.50e7i)T^{2} \)
73 \( 1 + (258. - 1.46e3i)T + (-2.66e7 - 9.71e6i)T^{2} \)
79 \( 1 + (-8.75e3 - 1.54e3i)T + (3.66e7 + 1.33e7i)T^{2} \)
83 \( 1 + (-3.71e3 - 6.44e3i)T + (-2.37e7 + 4.11e7i)T^{2} \)
89 \( 1 + (4.04e3 - 712. i)T + (5.89e7 - 2.14e7i)T^{2} \)
97 \( 1 + (2.46e3 + 6.77e3i)T + (-6.78e7 + 5.69e7i)T^{2} \)
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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

−13.39787860880583836538737933393, −12.57658853452028452241508389347, −11.87513818638315310376421928607, −10.65820533241839258589377358921, −9.923349895331728224087853796110, −7.59723636663417624761417396937, −6.86873302148017017891169677116, −5.33211403022657548867957869665, −4.33353993277139490188600768409, −1.00265052177866970043513434296, 0.37686698921318365416916284022, 3.53755564761111289891208541145, 5.44567366938827743391606146693, 6.02097845567805496531838450534, 7.48562783563535090106268277365, 9.355225053432475660462657186869, 10.75544852239902807110224407946, 11.29106057572985082936237042115, 12.20183486432766955890359132799, 13.27687621704276895577367326763

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