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Properties

Label 2-78-39.20-c1-0-2
Degree $2$
Conductor $78$
Sign $0.877 - 0.480i$
Analytic cond. $0.622833$
Root an. cond. $0.789197$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.258 + 0.965i)2-s + (1.45 − 0.933i)3-s + (−0.866 + 0.499i)4-s + (−0.428 + 0.428i)5-s + (1.27 + 1.16i)6-s + (−0.735 − 0.196i)7-s + (−0.707 − 0.707i)8-s + (1.25 − 2.72i)9-s + (−0.524 − 0.303i)10-s + (−4.05 + 1.08i)11-s + (−0.796 + 1.53i)12-s + (0.601 + 3.55i)13-s − 0.761i·14-s + (−0.224 + 1.02i)15-s + (0.500 − 0.866i)16-s + (−2.62 − 4.54i)17-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (0.842 − 0.539i)3-s + (−0.433 + 0.249i)4-s + (−0.191 + 0.191i)5-s + (0.522 + 0.476i)6-s + (−0.277 − 0.0744i)7-s + (−0.249 − 0.249i)8-s + (0.418 − 0.908i)9-s + (−0.165 − 0.0958i)10-s + (−1.22 + 0.327i)11-s + (−0.229 + 0.444i)12-s + (0.166 + 0.986i)13-s − 0.203i·14-s + (−0.0580 + 0.264i)15-s + (0.125 − 0.216i)16-s + (−0.636 − 1.10i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.877 - 0.480i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.877 - 0.480i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(78\)    =    \(2 \cdot 3 \cdot 13\)
Sign: $0.877 - 0.480i$
Analytic conductor: \(0.622833\)
Root analytic conductor: \(0.789197\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{78} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 78,\ (\ :1/2),\ 0.877 - 0.480i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10737 + 0.283381i\)
\(L(\frac12)\) \(\approx\) \(1.10737 + 0.283381i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 - 0.965i)T \)
3 \( 1 + (-1.45 + 0.933i)T \)
13 \( 1 + (-0.601 - 3.55i)T \)
good5 \( 1 + (0.428 - 0.428i)T - 5iT^{2} \)
7 \( 1 + (0.735 + 0.196i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (4.05 - 1.08i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (2.62 + 4.54i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.882 + 3.29i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-0.933 + 1.61i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-7.53 - 4.35i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.68 + 2.68i)T + 31iT^{2} \)
37 \( 1 + (-1.52 - 5.67i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-2.29 - 8.56i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-1.68 + 0.975i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.73 - 5.73i)T + 47iT^{2} \)
53 \( 1 + 9.01iT - 53T^{2} \)
59 \( 1 + (-2.23 + 8.34i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-4.06 - 7.04i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.101 - 0.0271i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (10.0 + 2.69i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-5.57 + 5.57i)T - 73iT^{2} \)
79 \( 1 + 13.5T + 79T^{2} \)
83 \( 1 + (-0.996 + 0.996i)T - 83iT^{2} \)
89 \( 1 + (6.32 - 1.69i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-4.07 + 15.2i)T + (-84.0 - 48.5i)T^{2} \)
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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

−14.51070008739830812122573070843, −13.52678326174195487511776447573, −12.86225127019444139719110933035, −11.44158800207031341369988326434, −9.721221701371409634103174301677, −8.682869364172419374794005178184, −7.44508159921891666088042608864, −6.65753392389661077644122146085, −4.71781976146087273352303678615, −2.87612980037480533274028278029, 2.67624486337969997461794688461, 4.06253920901433811902486824669, 5.59933010440160890229430976757, 7.889213641449688999845529884395, 8.744644352014627120117458838396, 10.19540028388725989548926704064, 10.71920235471392624735159681340, 12.44620051228390867603081973910, 13.22984654472457049492835356663, 14.22766218476589071952346534602

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