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Properties

Label 2-78-13.7-c2-0-1
Degree $2$
Conductor $78$
Sign $0.895 + 0.445i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.36 − 0.366i)2-s + (0.866 − 1.5i)3-s + (1.73 − i)4-s + (4.02 + 4.02i)5-s + (0.633 − 2.36i)6-s + (−4.99 − 1.33i)7-s + (1.99 − 2i)8-s + (−1.5 − 2.59i)9-s + (6.97 + 4.02i)10-s + (−3.41 − 12.7i)11-s − 3.46i·12-s + (3.81 + 12.4i)13-s − 7.31·14-s + (9.52 − 2.55i)15-s + (1.99 − 3.46i)16-s + (−16.3 + 9.42i)17-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (0.288 − 0.5i)3-s + (0.433 − 0.250i)4-s + (0.805 + 0.805i)5-s + (0.105 − 0.394i)6-s + (−0.714 − 0.191i)7-s + (0.249 − 0.250i)8-s + (−0.166 − 0.288i)9-s + (0.697 + 0.402i)10-s + (−0.310 − 1.15i)11-s − 0.288i·12-s + (0.293 + 0.956i)13-s − 0.522·14-s + (0.635 − 0.170i)15-s + (0.124 − 0.216i)16-s + (−0.960 + 0.554i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 + 0.445i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.895 + 0.445i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(78\)    =    \(2 \cdot 3 \cdot 13\)
Sign: $0.895 + 0.445i$
Analytic conductor: \(2.12534\)
Root analytic conductor: \(1.45785\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{78} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 78,\ (\ :1),\ 0.895 + 0.445i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.93001 - 0.453849i\)
\(L(\frac12)\) \(\approx\) \(1.93001 - 0.453849i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36 + 0.366i)T \)
3 \( 1 + (-0.866 + 1.5i)T \)
13 \( 1 + (-3.81 - 12.4i)T \)
good5 \( 1 + (-4.02 - 4.02i)T + 25iT^{2} \)
7 \( 1 + (4.99 + 1.33i)T + (42.4 + 24.5i)T^{2} \)
11 \( 1 + (3.41 + 12.7i)T + (-104. + 60.5i)T^{2} \)
17 \( 1 + (16.3 - 9.42i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (7.85 - 29.3i)T + (-312. - 180.5i)T^{2} \)
23 \( 1 + (10.4 + 6.05i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-18.4 + 31.9i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (6.48 + 6.48i)T + 961iT^{2} \)
37 \( 1 + (-13.6 - 50.7i)T + (-1.18e3 + 684.5i)T^{2} \)
41 \( 1 + (6.38 - 1.70i)T + (1.45e3 - 840.5i)T^{2} \)
43 \( 1 + (-11.7 + 6.80i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-61.6 + 61.6i)T - 2.20e3iT^{2} \)
53 \( 1 - 4.64T + 2.80e3T^{2} \)
59 \( 1 + (-24.1 - 6.45i)T + (3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (19.2 + 33.3i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-91.8 + 24.6i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + (-10.5 + 39.2i)T + (-4.36e3 - 2.52e3i)T^{2} \)
73 \( 1 + (-60.4 + 60.4i)T - 5.32e3iT^{2} \)
79 \( 1 + 94.2T + 6.24e3T^{2} \)
83 \( 1 + (63.4 + 63.4i)T + 6.88e3iT^{2} \)
89 \( 1 + (-40.3 - 150. i)T + (-6.85e3 + 3.96e3i)T^{2} \)
97 \( 1 + (39.2 - 146. i)T + (-8.14e3 - 4.70e3i)T^{2} \)
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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

−13.77970767469074200095028040316, −13.45267820216474587961072514927, −12.12049479701729260829120512037, −10.86763285361494331761247741347, −9.892924643009544628558941781215, −8.328557056663830287523161360047, −6.55011400135775077158026706076, −6.08742647797107526928395949237, −3.75848604698402290349650874034, −2.26404979982670582837484680998, 2.60663425455804175641434139017, 4.52559726024657033911490971382, 5.58224358065505962444410043928, 7.07945159541479796051819060587, 8.806611844648295471473901094277, 9.699519629941042810164147850321, 10.95081036181626650825447065815, 12.70188107472452708306478363051, 13.03239388779931940198648714943, 14.19669188061488014068221775101

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