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Properties

Label 2-78-13.4-c1-0-2
Degree $2$
Conductor $78$
Sign $0.454 + 0.890i$
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 − 1.73i·5-s + (−0.866 + 0.499i)6-s + (1.09 − 0.633i)7-s − 0.999i·8-s + (−0.499 − 0.866i)9-s + (−0.866 + 1.49i)10-s + (−1.09 − 0.633i)11-s + 0.999·12-s + (1.59 + 3.23i)13-s − 1.26·14-s + (−1.49 − 0.866i)15-s + (−0.5 + 0.866i)16-s + (2.59 + 4.5i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.288 − 0.499i)3-s + (0.249 + 0.433i)4-s − 0.774i·5-s + (−0.353 + 0.204i)6-s + (0.415 − 0.239i)7-s − 0.353i·8-s + (−0.166 − 0.288i)9-s + (−0.273 + 0.474i)10-s + (−0.331 − 0.191i)11-s + 0.288·12-s + (0.443 + 0.896i)13-s − 0.338·14-s + (−0.387 − 0.223i)15-s + (−0.125 + 0.216i)16-s + (0.630 + 1.09i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.454 + 0.890i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.454 + 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(78\)    =    \(2 \cdot 3 \cdot 13\)
Sign: $0.454 + 0.890i$
Analytic conductor: \(0.622833\)
Root analytic conductor: \(0.789197\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{78} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 78,\ (\ :1/2),\ 0.454 + 0.890i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.681572 - 0.417303i\)
\(L(\frac12)\) \(\approx\) \(0.681572 - 0.417303i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.59 - 3.23i)T \)
good5 \( 1 + 1.73iT - 5T^{2} \)
7 \( 1 + (-1.09 + 0.633i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.09 + 0.633i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.59 - 4.5i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.09 - 2.36i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.09 - 7.09i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 9.46iT - 31T^{2} \)
37 \( 1 + (2.59 + 1.5i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.59 - 3.23i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.09 + 3.63i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 4.73iT - 47T^{2} \)
53 \( 1 - 3T + 53T^{2} \)
59 \( 1 + (12 - 6.92i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.59 + 13.1i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.29 - 3.63i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.90 + 1.09i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 12.1iT - 73T^{2} \)
79 \( 1 - 8.39T + 79T^{2} \)
83 \( 1 + 5.66iT - 83T^{2} \)
89 \( 1 + (8.19 + 4.73i)T + (44.5 + 77.0i)T^{2} \)
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

−14.07438135183436592016473242305, −13.04661711034216813681299315316, −12.10356732019519162276869562380, −11.00811460088405517289531738082, −9.664680484433959991954181692700, −8.489836018754490918413313919452, −7.73448658827707220274435635388, −6.03580985518465871879523139507, −4.01113955825400049886541738745, −1.70157047061208495340155103517, 2.83643051103933429250281585704, 4.98937760909974793030923450180, 6.56786227303523526596374036781, 7.904887659821208145754925150705, 8.917102257824684048205035529725, 10.33620147586997298665821387731, 10.85202199386372656451985597941, 12.39034252874743331753886271598, 14.02294098635372501186311735128, 14.77057309090860548221245461989

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