Properties

Label 2-78-13.11-c2-0-0
Degree $2$
Conductor $78$
Sign $-0.295 - 0.955i$
Analytic cond. $2.12534$
Root an. cond. $1.45785$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.366 + 1.36i)2-s + (0.866 + 1.5i)3-s + (−1.73 − i)4-s + (0.633 + 0.633i)5-s + (−2.36 + 0.633i)6-s + (3.26 + 12.1i)7-s + (2 − 1.99i)8-s + (−1.5 + 2.59i)9-s + (−1.09 + 0.633i)10-s + (−17.6 − 4.73i)11-s − 3.46i·12-s + (11.2 − 6.5i)13-s − 17.8·14-s + (−0.401 + 1.5i)15-s + (1.99 + 3.46i)16-s + (15.1 + 8.76i)17-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (0.288 + 0.5i)3-s + (−0.433 − 0.250i)4-s + (0.126 + 0.126i)5-s + (−0.394 + 0.105i)6-s + (0.466 + 1.74i)7-s + (0.250 − 0.249i)8-s + (−0.166 + 0.288i)9-s + (−0.109 + 0.0633i)10-s + (−1.60 − 0.430i)11-s − 0.288i·12-s + (0.866 − 0.5i)13-s − 1.27·14-s + (−0.0267 + 0.100i)15-s + (0.124 + 0.216i)16-s + (0.893 + 0.515i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.295 - 0.955i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.295 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(78\)    =    \(2 \cdot 3 \cdot 13\)
Sign: $-0.295 - 0.955i$
Analytic conductor: \(2.12534\)
Root analytic conductor: \(1.45785\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{78} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 78,\ (\ :1),\ -0.295 - 0.955i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.721665 + 0.979065i\)
\(L(\frac12)\) \(\approx\) \(0.721665 + 0.979065i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.366 - 1.36i)T \)
3 \( 1 + (-0.866 - 1.5i)T \)
13 \( 1 + (-11.2 + 6.5i)T \)
good5 \( 1 + (-0.633 - 0.633i)T + 25iT^{2} \)
7 \( 1 + (-3.26 - 12.1i)T + (-42.4 + 24.5i)T^{2} \)
11 \( 1 + (17.6 + 4.73i)T + (104. + 60.5i)T^{2} \)
17 \( 1 + (-15.1 - 8.76i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-16.1 + 4.33i)T + (312. - 180.5i)T^{2} \)
23 \( 1 + (-18.5 + 10.7i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (19.3 + 33.4i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (-25.4 - 25.4i)T + 961iT^{2} \)
37 \( 1 + (-2.96 - 0.794i)T + (1.18e3 + 684.5i)T^{2} \)
41 \( 1 + (-7.91 + 29.5i)T + (-1.45e3 - 840.5i)T^{2} \)
43 \( 1 + (15.3 + 8.87i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (30 - 30i)T - 2.20e3iT^{2} \)
53 \( 1 + 23.1T + 2.80e3T^{2} \)
59 \( 1 + (-5.50 - 20.5i)T + (-3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (-21.1 + 36.5i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (8.48 - 31.6i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + (-33.1 + 8.87i)T + (4.36e3 - 2.52e3i)T^{2} \)
73 \( 1 + (15.9 - 15.9i)T - 5.32e3iT^{2} \)
79 \( 1 + 89.5T + 6.24e3T^{2} \)
83 \( 1 + (-97.4 - 97.4i)T + 6.88e3iT^{2} \)
89 \( 1 + (-1.22 - 0.327i)T + (6.85e3 + 3.96e3i)T^{2} \)
97 \( 1 + (49.3 - 13.2i)T + (8.14e3 - 4.70e3i)T^{2} \)
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

−14.84433444169947298914040829288, −13.67588190926095216768974632583, −12.49333067477499206979336944350, −11.05256190097259277107042518994, −9.896193995118790347644771205747, −8.578708701876584785590095623752, −8.012495736319474737216201064691, −5.90845196077048521431817502011, −5.15472722294144304002350835749, −2.84553333759888986071862184326, 1.26231715631372448872146526690, 3.38931376753971293508525940402, 5.05133854277541149603280165391, 7.26950290388799985155144269871, 7.960563421026368266781007428204, 9.586440370901569575103216749606, 10.61895277601827403487264880838, 11.53982277250389510112235440601, 13.15014508978659364390962524704, 13.45234724026599152183050515191

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