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Properties

Label 2-78-13.8-c2-0-2
Degree $2$
Conductor $78$
Sign $0.898 + 0.438i$
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 + (3.96 + 3.96i)5-s + (1.73 + 1.73i)6-s + (8.11 − 8.11i)7-s + (2 − 2i)8-s + 2.99·9-s − 7.93i·10-s + (0.0403 − 0.0403i)11-s − 3.46i·12-s + (10.5 − 7.60i)13-s − 16.2·14-s + (−6.87 − 6.87i)15-s − 4·16-s + 25.1i·17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s − 0.577·3-s + 0.5i·4-s + (0.793 + 0.793i)5-s + (0.288 + 0.288i)6-s + (1.15 − 1.15i)7-s + (0.250 − 0.250i)8-s + 0.333·9-s − 0.793i·10-s + (0.00366 − 0.00366i)11-s − 0.288i·12-s + (0.810 − 0.585i)13-s − 1.15·14-s + (−0.458 − 0.458i)15-s − 0.250·16-s + 1.47i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.898 + 0.438i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.898 + 0.438i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(78\)    =    \(2 \cdot 3 \cdot 13\)
Sign: $0.898 + 0.438i$
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: \((2,\ 78,\ (\ :1),\ 0.898 + 0.438i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.01704 - 0.235156i\)
\(L(\frac12)\) \(\approx\) \(1.01704 - 0.235156i\)
\(L(2)\) 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 \( 1 + (-10.5 + 7.60i)T \)
good5 \( 1 + (-3.96 - 3.96i)T + 25iT^{2} \)
7 \( 1 + (-8.11 + 8.11i)T - 49iT^{2} \)
11 \( 1 + (-0.0403 + 0.0403i)T - 121iT^{2} \)
17 \( 1 - 25.1iT - 289T^{2} \)
19 \( 1 + (-13.1 - 13.1i)T + 361iT^{2} \)
23 \( 1 + 18.8iT - 529T^{2} \)
29 \( 1 + 36.5T + 841T^{2} \)
31 \( 1 + (22.0 + 22.0i)T + 961iT^{2} \)
37 \( 1 + (45.9 - 45.9i)T - 1.36e3iT^{2} \)
41 \( 1 + (11.1 + 11.1i)T + 1.68e3iT^{2} \)
43 \( 1 + 20.9iT - 1.84e3T^{2} \)
47 \( 1 + (-12.8 + 12.8i)T - 2.20e3iT^{2} \)
53 \( 1 - 65.0T + 2.80e3T^{2} \)
59 \( 1 + (51.8 - 51.8i)T - 3.48e3iT^{2} \)
61 \( 1 - 44.0T + 3.72e3T^{2} \)
67 \( 1 + (-38.0 - 38.0i)T + 4.48e3iT^{2} \)
71 \( 1 + (67.2 + 67.2i)T + 5.04e3iT^{2} \)
73 \( 1 + (97.6 - 97.6i)T - 5.32e3iT^{2} \)
79 \( 1 + 82.7T + 6.24e3T^{2} \)
83 \( 1 + (-17.4 - 17.4i)T + 6.88e3iT^{2} \)
89 \( 1 + (3.53 - 3.53i)T - 7.92e3iT^{2} \)
97 \( 1 + (5.98 + 5.98i)T + 9.40e3iT^{2} \)
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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

−14.00685699689619368038694952676, −13.03741430791585871941156782611, −11.57116761501365539736858942201, −10.52526157721302395876047874681, −10.30882109893081070455937696338, −8.396116429225048290219798148950, −7.20009770479894773399381857426, −5.76748123729880018213517158878, −3.88589091918768698246072662002, −1.57100683833812999620357712852, 1.64458193718231851368893385190, 5.08151271056715145792851226004, 5.63579062968254320190766781810, 7.30829661591784271849138450517, 8.873781705902084319147924798009, 9.362243464909108140274871379800, 11.11293081123121794305070466174, 11.86781848355568513219378560796, 13.31555124238673393424105446187, 14.33418174537896483157421415192

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