Properties

Label 2-78-13.5-c2-0-3
Degree $2$
Conductor $78$
Sign $-0.670 + 0.741i$
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 + (−2.23 + 2.23i)5-s + (1.73 − 1.73i)6-s + (−8.84 − 8.84i)7-s + (2 + 2i)8-s + 2.99·9-s − 4.47i·10-s + (−6.16 − 6.16i)11-s + 3.46i·12-s + (−12.6 − 3.14i)13-s + 17.6·14-s + (3.87 − 3.87i)15-s − 4·16-s + 8.75i·17-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s − 0.577·3-s − 0.5i·4-s + (−0.447 + 0.447i)5-s + (0.288 − 0.288i)6-s + (−1.26 − 1.26i)7-s + (0.250 + 0.250i)8-s + 0.333·9-s − 0.447i·10-s + (−0.560 − 0.560i)11-s + 0.288i·12-s + (−0.970 − 0.241i)13-s + 1.26·14-s + (0.258 − 0.258i)15-s − 0.250·16-s + 0.515i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.670 + 0.741i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.670 + 0.741i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(78\)    =    \(2 \cdot 3 \cdot 13\)
Sign: $-0.670 + 0.741i$
Analytic conductor: \(2.12534\)
Root analytic conductor: \(1.45785\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{78} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 78,\ (\ :1),\ -0.670 + 0.741i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0647446 - 0.145898i\)
\(L(\frac12)\) \(\approx\) \(0.0647446 - 0.145898i\)
\(L(2)\) 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 \( 1 + (12.6 + 3.14i)T \)
good5 \( 1 + (2.23 - 2.23i)T - 25iT^{2} \)
7 \( 1 + (8.84 + 8.84i)T + 49iT^{2} \)
11 \( 1 + (6.16 + 6.16i)T + 121iT^{2} \)
17 \( 1 - 8.75iT - 289T^{2} \)
19 \( 1 + (-8.55 + 8.55i)T - 361iT^{2} \)
23 \( 1 - 31.2iT - 529T^{2} \)
29 \( 1 + 33.1T + 841T^{2} \)
31 \( 1 + (-7.31 + 7.31i)T - 961iT^{2} \)
37 \( 1 + (-0.372 - 0.372i)T + 1.36e3iT^{2} \)
41 \( 1 + (-41.3 + 41.3i)T - 1.68e3iT^{2} \)
43 \( 1 + 80.7iT - 1.84e3T^{2} \)
47 \( 1 + (42.9 + 42.9i)T + 2.20e3iT^{2} \)
53 \( 1 + 70.5T + 2.80e3T^{2} \)
59 \( 1 + (24.1 + 24.1i)T + 3.48e3iT^{2} \)
61 \( 1 - 90.3T + 3.72e3T^{2} \)
67 \( 1 + (37.6 - 37.6i)T - 4.48e3iT^{2} \)
71 \( 1 + (14.7 - 14.7i)T - 5.04e3iT^{2} \)
73 \( 1 + (14.0 + 14.0i)T + 5.32e3iT^{2} \)
79 \( 1 - 59.5T + 6.24e3T^{2} \)
83 \( 1 + (35.1 - 35.1i)T - 6.88e3iT^{2} \)
89 \( 1 + (-11.7 - 11.7i)T + 7.92e3iT^{2} \)
97 \( 1 + (-3.09 + 3.09i)T - 9.40e3iT^{2} \)
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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

−13.77003520949083552773285584864, −12.86171824849266876373489568801, −11.31226108475634532467825613626, −10.38639244100834291389235794694, −9.501629341996498792425806726913, −7.58820051897236532328398918033, −6.97639486561429755814366699598, −5.54588773127492989120239807667, −3.59934186764951318589897526157, −0.15988640227079184540814542011, 2.70217246716142205545733403495, 4.77173022101606831358097561712, 6.34377995043752806978345062387, 7.83379131215489550633610221991, 9.282474838907772617380220276865, 9.975784054713016111659273513358, 11.49023582501835350154068401426, 12.48135078970438337814941638243, 12.77532733804999172742329193640, 14.78956599686875456724820903499

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