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Properties

Label 2-78-13.10-c1-0-3
Degree $2$
Conductor $78$
Sign $0.711 + 0.702i$
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  + (0.866 − 0.5i)2-s + (−0.5 − 0.866i)3-s + (0.499 − 0.866i)4-s + 0.267i·5-s + (−0.866 − 0.499i)6-s + (0.633 + 0.366i)7-s − 0.999i·8-s + (−0.499 + 0.866i)9-s + (0.133 + 0.232i)10-s + (−4.09 + 2.36i)11-s − 0.999·12-s + (2.59 + 2.5i)13-s + 0.732·14-s + (0.232 − 0.133i)15-s + (−0.5 − 0.866i)16-s + (−1.13 + 1.96i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.288 − 0.499i)3-s + (0.249 − 0.433i)4-s + 0.119i·5-s + (−0.353 − 0.204i)6-s + (0.239 + 0.138i)7-s − 0.353i·8-s + (−0.166 + 0.288i)9-s + (0.0423 + 0.0733i)10-s + (−1.23 + 0.713i)11-s − 0.288·12-s + (0.720 + 0.693i)13-s + 0.195·14-s + (0.0599 − 0.0345i)15-s + (−0.125 − 0.216i)16-s + (−0.275 + 0.476i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(78\)    =    \(2 \cdot 3 \cdot 13\)
Sign: $0.711 + 0.702i$
Analytic conductor: \(0.622833\)
Root analytic conductor: \(0.789197\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{78} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 78,\ (\ :1/2),\ 0.711 + 0.702i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06959 - 0.439029i\)
\(L(\frac12)\) \(\approx\) \(1.06959 - 0.439029i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (-2.59 - 2.5i)T \)
good5 \( 1 - 0.267iT - 5T^{2} \)
7 \( 1 + (-0.633 - 0.366i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (4.09 - 2.36i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.13 - 1.96i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.09 + 0.633i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.09 + 5.36i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.23 + 2.13i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 5.46iT - 31T^{2} \)
37 \( 1 + (9.06 - 5.23i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-9.86 + 5.69i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.83 + 6.63i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 8.19iT - 47T^{2} \)
53 \( 1 - 0.464T + 53T^{2} \)
59 \( 1 + (-6.92 - 4i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.598 - 1.03i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (9.63 - 5.56i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.09 + 0.633i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 9.73iT - 73T^{2} \)
79 \( 1 + 9.46T + 79T^{2} \)
83 \( 1 - 10.1iT - 83T^{2} \)
89 \( 1 + (-2.19 + 1.26i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.19 - 3i)T + (48.5 + 84.0i)T^{2} \)
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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

−14.13780100489033288768875557517, −13.09688708779337892690664683882, −12.33620932624610486368410457367, −11.14920526402552747920383678637, −10.27621947446913151759042079185, −8.549207448099631054207377829777, −7.11253044695813313555695908315, −5.84002869441533288882442948255, −4.41295548574965369783604137822, −2.28101002280442087926870223077, 3.29459443722306480203783628955, 4.94944937197601151130105246391, 5.94132277755358982544269229002, 7.60494649373673797247094018464, 8.788196249468763156294424808134, 10.46952276357695368955839763573, 11.23685771339655864242991003883, 12.63197640964496097825873655354, 13.54569331498988645293252872179, 14.60577775424719919520870272926

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