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Properties

Label 2-76-4.3-c4-0-35
Degree $2$
Conductor $76$
Sign $-0.331 - 0.943i$
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  + (−0.672 − 3.94i)2-s − 15.5i·3-s + (−15.0 + 5.30i)4-s − 5.44·5-s + (−61.2 + 10.4i)6-s − 31.1i·7-s + (31.0 + 55.9i)8-s − 160.·9-s + (3.66 + 21.4i)10-s − 45.9i·11-s + (82.3 + 234. i)12-s + 7.06·13-s + (−122. + 20.9i)14-s + 84.6i·15-s + (199. − 160. i)16-s + 382.·17-s + ⋯
L(s)  = 1  + (−0.168 − 0.985i)2-s − 1.72i·3-s + (−0.943 + 0.331i)4-s − 0.217·5-s + (−1.70 + 0.290i)6-s − 0.636i·7-s + (0.485 + 0.874i)8-s − 1.97·9-s + (0.0366 + 0.214i)10-s − 0.379i·11-s + (0.571 + 1.62i)12-s + 0.0418·13-s + (−0.627 + 0.106i)14-s + 0.376i·15-s + (0.780 − 0.625i)16-s + 1.32·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.331 - 0.943i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.331 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-0.331 - 0.943i$
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: \((2,\ 76,\ (\ :2),\ -0.331 - 0.943i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.504999 + 0.712533i\)
\(L(\frac12)\) \(\approx\) \(0.504999 + 0.712533i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.672 + 3.94i)T \)
19 \( 1 - 82.8iT \)
good3 \( 1 + 15.5iT - 81T^{2} \)
5 \( 1 + 5.44T + 625T^{2} \)
7 \( 1 + 31.1iT - 2.40e3T^{2} \)
11 \( 1 + 45.9iT - 1.46e4T^{2} \)
13 \( 1 - 7.06T + 2.85e4T^{2} \)
17 \( 1 - 382.T + 8.35e4T^{2} \)
23 \( 1 - 212. iT - 2.79e5T^{2} \)
29 \( 1 + 1.48e3T + 7.07e5T^{2} \)
31 \( 1 + 1.11e3iT - 9.23e5T^{2} \)
37 \( 1 - 720.T + 1.87e6T^{2} \)
41 \( 1 + 679.T + 2.82e6T^{2} \)
43 \( 1 - 1.51e3iT - 3.41e6T^{2} \)
47 \( 1 + 3.91e3iT - 4.87e6T^{2} \)
53 \( 1 + 3.91e3T + 7.89e6T^{2} \)
59 \( 1 + 5.16e3iT - 1.21e7T^{2} \)
61 \( 1 + 1.97e3T + 1.38e7T^{2} \)
67 \( 1 - 2.69e3iT - 2.01e7T^{2} \)
71 \( 1 - 7.27e3iT - 2.54e7T^{2} \)
73 \( 1 - 8.38e3T + 2.83e7T^{2} \)
79 \( 1 + 3.24e3iT - 3.89e7T^{2} \)
83 \( 1 + 1.16e4iT - 4.74e7T^{2} \)
89 \( 1 - 1.33e4T + 6.27e7T^{2} \)
97 \( 1 + 3.76e3T + 8.85e7T^{2} \)
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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

−12.97170344906396377606840731997, −11.93562545492456759828917377112, −11.19410955514954754584556258952, −9.683436299985974091889873885423, −8.109019827176976846235153886097, −7.47115962503711195347999941621, −5.75317744472651644192049101912, −3.52875563937694210620170603185, −1.80756790663461143263913944069, −0.48712048282062963157502167784, 3.61415811960795028567953829704, 4.89694660206687294264505971202, 5.86934295346084983021614929355, 7.77566637584079887450555457679, 9.049181537784975768217210394689, 9.737099677547786242168451869056, 10.83139640711480433877151377467, 12.32669736407524286556196980112, 14.02711831121069231057811733330, 14.95346698864840057643929343148

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