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Properties

Label 2-78-13.3-c3-0-7
Degree $2$
Conductor $78$
Sign $-0.380 + 0.924i$
Analytic cond. $4.60214$
Root an. cond. $2.14526$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 + 1.73i)2-s + (−1.5 − 2.59i)3-s + (−1.99 + 3.46i)4-s − 19.0·5-s + (3 − 5.19i)6-s + (16.8 − 29.2i)7-s − 7.99·8-s + (−4.5 + 7.79i)9-s + (−19.0 − 32.9i)10-s + (−29.9 − 51.8i)11-s + 12·12-s + (−28.6 + 37.0i)13-s + 67.5·14-s + (28.5 + 49.4i)15-s + (−8 − 13.8i)16-s + (−18.6 + 32.3i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s − 1.70·5-s + (0.204 − 0.353i)6-s + (0.911 − 1.57i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.601 − 1.04i)10-s + (−0.820 − 1.42i)11-s + 0.288·12-s + (−0.612 + 0.790i)13-s + 1.28·14-s + (0.491 + 0.851i)15-s + (−0.125 − 0.216i)16-s + (−0.266 + 0.461i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.380 + 0.924i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.380 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(78\)    =    \(2 \cdot 3 \cdot 13\)
Sign: $-0.380 + 0.924i$
Analytic conductor: \(4.60214\)
Root analytic conductor: \(2.14526\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{78} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 78,\ (\ :3/2),\ -0.380 + 0.924i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.358155 - 0.534780i\)
\(L(\frac12)\) \(\approx\) \(0.358155 - 0.534780i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - 1.73i)T \)
3 \( 1 + (1.5 + 2.59i)T \)
13 \( 1 + (28.6 - 37.0i)T \)
good5 \( 1 + 19.0T + 125T^{2} \)
7 \( 1 + (-16.8 + 29.2i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (29.9 + 51.8i)T + (-665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (18.6 - 32.3i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (5.57 - 9.65i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (25.8 + 44.6i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-24.0 - 41.6i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 66.8T + 2.97e4T^{2} \)
37 \( 1 + (112. + 194. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-54.1 - 93.7i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-4.88 + 8.45i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 119.T + 1.03e5T^{2} \)
53 \( 1 + 466.T + 1.48e5T^{2} \)
59 \( 1 + (-360. + 624. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-298. + 517. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (182. + 315. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-133. + 232. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 - 801.T + 3.89e5T^{2} \)
79 \( 1 + 931.T + 4.93e5T^{2} \)
83 \( 1 + 179.T + 5.71e5T^{2} \)
89 \( 1 + (-324. - 561. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (401. - 695. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−13.80715705312052452800992945847, −12.58635447455860502583665358181, −11.39040282128385730836977115379, −10.79941602887790775639312597032, −8.294050483963898984456555291185, −7.77059228565436095792956752159, −6.77187642310984906057446515488, −4.83258721643867679117138764658, −3.75240477907389778575134683963, −0.37107855118544775328536758729, 2.67029999731273075566347541965, 4.48516292211157082745253137782, 5.26989468159640671110106129419, 7.54073871365210406315700151294, 8.626299926125838150615392781464, 10.08259426794993485342218853216, 11.36187682507713207406895432561, 11.98789071076257627535491303821, 12.65479415005825375893673765025, 14.69510856411660149579366568549

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