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Properties

Label 2-78-39.2-c1-0-0
Degree $2$
Conductor $78$
Sign $-0.114 - 0.993i$
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 + (0.0795 + 1.73i)3-s + (−0.866 − 0.499i)4-s + (0.428 + 0.428i)5-s + (−1.69 − 0.370i)6-s + (−0.735 + 0.196i)7-s + (0.707 − 0.707i)8-s + (−2.98 + 0.275i)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 + (−0.707 + 0.775i)15-s + (0.500 + 0.866i)16-s + (2.62 − 4.54i)17-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (0.0459 + 0.998i)3-s + (−0.433 − 0.249i)4-s + (0.191 + 0.191i)5-s + (−0.690 − 0.151i)6-s + (−0.277 + 0.0744i)7-s + (0.249 − 0.249i)8-s + (−0.995 + 0.0917i)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 + (−0.182 + 0.200i)15-s + (0.125 + 0.216i)16-s + (0.636 − 1.10i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.114 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.114 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(78\)    =    \(2 \cdot 3 \cdot 13\)
Sign: $-0.114 - 0.993i$
Analytic conductor: \(0.622833\)
Root analytic conductor: \(0.789197\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{78} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 78,\ (\ :1/2),\ -0.114 - 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.574564 + 0.644377i\)
\(L(\frac12)\) \(\approx\) \(0.574564 + 0.644377i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0795 - 1.73i)T \)
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

−14.59576462229889452394170913585, −14.33001499173819825960684113335, −12.65986385575976139021334033807, −11.26243600067294551120394109531, −9.963501532349314672486407098614, −9.301128984402719030244376372178, −7.951190252797379421512124091651, −6.37820965575078920695623853633, −5.14371812008972933729227682288, −3.53331573613892207298866213562, 1.65475388515558011554016910091, 3.70664946501204209056340130831, 5.89581630389710450601912027967, 7.19816117745753386736897224030, 8.657712922077505302426066073443, 9.537693302174558347444527537782, 11.23018918761832898011857410046, 11.92423826064731441019500552773, 13.06574113014327687938441227881, 13.79983977459864300839528534032

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