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Properties

Label 2-78-13.5-c2-0-4
Degree $2$
Conductor $78$
Sign $0.697 + 0.716i$
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 + (1.26 − 1.26i)5-s + (1.73 − 1.73i)6-s + (0.732 + 0.732i)7-s + (−2 − 2i)8-s + 2.99·9-s − 2.53i·10-s + (−1.73 − 1.73i)11-s − 3.46i·12-s + (−3.92 + 12.3i)13-s + 1.46·14-s + (2.19 − 2.19i)15-s − 4·16-s − 5.32i·17-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s + 0.577·3-s − 0.5i·4-s + (0.253 − 0.253i)5-s + (0.288 − 0.288i)6-s + (0.104 + 0.104i)7-s + (−0.250 − 0.250i)8-s + 0.333·9-s − 0.253i·10-s + (−0.157 − 0.157i)11-s − 0.288i·12-s + (−0.302 + 0.953i)13-s + 0.104·14-s + (0.146 − 0.146i)15-s − 0.250·16-s − 0.312i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.697 + 0.716i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.697 + 0.716i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(78\)    =    \(2 \cdot 3 \cdot 13\)
Sign: $0.697 + 0.716i$
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: \((2,\ 78,\ (\ :1),\ 0.697 + 0.716i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.72898 - 0.729337i\)
\(L(\frac12)\) \(\approx\) \(1.72898 - 0.729337i\)
\(L(2)\) 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 \( 1 + (3.92 - 12.3i)T \)
good5 \( 1 + (-1.26 + 1.26i)T - 25iT^{2} \)
7 \( 1 + (-0.732 - 0.732i)T + 49iT^{2} \)
11 \( 1 + (1.73 + 1.73i)T + 121iT^{2} \)
17 \( 1 + 5.32iT - 289T^{2} \)
19 \( 1 + (14.7 - 14.7i)T - 361iT^{2} \)
23 \( 1 + 5.32iT - 529T^{2} \)
29 \( 1 - 4.14T + 841T^{2} \)
31 \( 1 + (24.9 - 24.9i)T - 961iT^{2} \)
37 \( 1 + (3.14 + 3.14i)T + 1.36e3iT^{2} \)
41 \( 1 + (-44.4 + 44.4i)T - 1.68e3iT^{2} \)
43 \( 1 + 37.1iT - 1.84e3T^{2} \)
47 \( 1 + (30.8 + 30.8i)T + 2.20e3iT^{2} \)
53 \( 1 - 57.7T + 2.80e3T^{2} \)
59 \( 1 + (66.6 + 66.6i)T + 3.48e3iT^{2} \)
61 \( 1 - 103.T + 3.72e3T^{2} \)
67 \( 1 + (-46.6 + 46.6i)T - 4.48e3iT^{2} \)
71 \( 1 + (26.9 - 26.9i)T - 5.04e3iT^{2} \)
73 \( 1 + (-5.67 - 5.67i)T + 5.32e3iT^{2} \)
79 \( 1 - 4.21T + 6.24e3T^{2} \)
83 \( 1 + (109. - 109. i)T - 6.88e3iT^{2} \)
89 \( 1 + (19.5 + 19.5i)T + 7.92e3iT^{2} \)
97 \( 1 + (-4.03 + 4.03i)T - 9.40e3iT^{2} \)
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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

−14.05022074096793938524701252267, −13.01751301021265366257630862930, −12.06996748201319764501955260071, −10.81249036430437921050728411986, −9.603373426463757757849281407219, −8.576448612925182911060043559716, −6.94724977497096726507135375243, −5.32042872923809144637685446922, −3.83585695595220724030054914464, −2.04895953306349928450982442824, 2.72332490700667382595560767124, 4.42686904735446228657874972521, 5.99592639545284492878394082543, 7.36612495223565123544504107311, 8.412123354644109433154813164313, 9.797433138008203317680747890569, 11.04572872376180467560621466891, 12.61032564693647504572864253688, 13.32582168937518093575634561378, 14.49772279980073101209607763738

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