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Properties

Label 2-78-13.8-c2-0-5
Degree $2$
Conductor $78$
Sign $-0.254 + 0.967i$
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 − i)2-s + 1.73·3-s + 2i·4-s + (−6.68 − 6.68i)5-s + (−1.73 − 1.73i)6-s + (5.62 − 5.62i)7-s + (2 − 2i)8-s + 2.99·9-s + 13.3i·10-s + (3.24 − 3.24i)11-s + 3.46i·12-s + (−9.52 − 8.84i)13-s − 11.2·14-s + (−11.5 − 11.5i)15-s − 4·16-s + 6.32i·17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + 0.577·3-s + 0.5i·4-s + (−1.33 − 1.33i)5-s + (−0.288 − 0.288i)6-s + (0.803 − 0.803i)7-s + (0.250 − 0.250i)8-s + 0.333·9-s + 1.33i·10-s + (0.294 − 0.294i)11-s + 0.288i·12-s + (−0.732 − 0.680i)13-s − 0.803·14-s + (−0.772 − 0.772i)15-s − 0.250·16-s + 0.372i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.254 + 0.967i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.254 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(78\)    =    \(2 \cdot 3 \cdot 13\)
Sign: $-0.254 + 0.967i$
Analytic conductor: \(2.12534\)
Root analytic conductor: \(1.45785\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{78} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 78,\ (\ :1),\ -0.254 + 0.967i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.602679 - 0.781768i\)
\(L(\frac12)\) \(\approx\) \(0.602679 - 0.781768i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + i)T \)
3 \( 1 - 1.73T \)
13 \( 1 + (9.52 + 8.84i)T \)
good5 \( 1 + (6.68 + 6.68i)T + 25iT^{2} \)
7 \( 1 + (-5.62 + 5.62i)T - 49iT^{2} \)
11 \( 1 + (-3.24 + 3.24i)T - 121iT^{2} \)
17 \( 1 - 6.32iT - 289T^{2} \)
19 \( 1 + (-18.0 - 18.0i)T + 361iT^{2} \)
23 \( 1 - 1.41iT - 529T^{2} \)
29 \( 1 - 36.2T + 841T^{2} \)
31 \( 1 + (-1.74 - 1.74i)T + 961iT^{2} \)
37 \( 1 + (-21.9 + 21.9i)T - 1.36e3iT^{2} \)
41 \( 1 + (-11.8 - 11.8i)T + 1.68e3iT^{2} \)
43 \( 1 + 61.4iT - 1.84e3T^{2} \)
47 \( 1 + (55.3 - 55.3i)T - 2.20e3iT^{2} \)
53 \( 1 - 72.8T + 2.80e3T^{2} \)
59 \( 1 + (-11.7 + 11.7i)T - 3.48e3iT^{2} \)
61 \( 1 + 82.0T + 3.72e3T^{2} \)
67 \( 1 + (39.7 + 39.7i)T + 4.48e3iT^{2} \)
71 \( 1 + (58.0 + 58.0i)T + 5.04e3iT^{2} \)
73 \( 1 + (-61.8 + 61.8i)T - 5.32e3iT^{2} \)
79 \( 1 - 124.T + 6.24e3T^{2} \)
83 \( 1 + (-35.9 - 35.9i)T + 6.88e3iT^{2} \)
89 \( 1 + (72.0 - 72.0i)T - 7.92e3iT^{2} \)
97 \( 1 + (-25.6 - 25.6i)T + 9.40e3iT^{2} \)
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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

−13.76892741595032316768474258976, −12.50965129209517827520910110705, −11.83442198271917063445122027102, −10.57669048814252196430836342784, −9.198870382916405338668957971453, −8.037901138911138668320886064888, −7.67224844531886941861943844651, −4.80013591728513582460883741637, −3.66440920370706673182360543887, −1.03310101060078596065428372742, 2.73178683302462964885393789170, 4.62045902462877960312607466184, 6.77701965075488696781838157621, 7.58399670744675440332280657462, 8.604686165480361857512758867650, 9.915522531714989302704369014928, 11.35347326380587830438738955383, 11.92002214100561607905270204119, 13.99389740613171241235074726806, 14.82335204450214246976896131210

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