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Properties

Label 2-78-39.5-c1-0-3
Degree $2$
Conductor $78$
Sign $0.649 + 0.760i$
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  + (−0.707 − 0.707i)2-s + (1.29 − 1.15i)3-s + 1.00i·4-s + (1.82 + 1.82i)5-s + (−1.72 − 0.0980i)6-s + (−2.63 − 2.63i)7-s + (0.707 − 0.707i)8-s + (0.339 − 2.98i)9-s − 2.58i·10-s + (−2.30 + 2.30i)11-s + (1.15 + 1.29i)12-s + (1.63 + 3.21i)13-s + 3.72i·14-s + (4.46 + 0.253i)15-s − 1.00·16-s − 1.34·17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (0.746 − 0.665i)3-s + 0.500i·4-s + (0.817 + 0.817i)5-s + (−0.705 − 0.0400i)6-s + (−0.994 − 0.994i)7-s + (0.250 − 0.250i)8-s + (0.113 − 0.993i)9-s − 0.817i·10-s + (−0.695 + 0.695i)11-s + (0.332 + 0.373i)12-s + (0.452 + 0.891i)13-s + 0.994i·14-s + (1.15 + 0.0654i)15-s − 0.250·16-s − 0.326·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.649 + 0.760i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.649 + 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(78\)    =    \(2 \cdot 3 \cdot 13\)
Sign: $0.649 + 0.760i$
Analytic conductor: \(0.622833\)
Root analytic conductor: \(0.789197\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{78} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 78,\ (\ :1/2),\ 0.649 + 0.760i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.843331 - 0.388790i\)
\(L(\frac12)\) \(\approx\) \(0.843331 - 0.388790i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
3 \( 1 + (-1.29 + 1.15i)T \)
13 \( 1 + (-1.63 - 3.21i)T \)
good5 \( 1 + (-1.82 - 1.82i)T + 5iT^{2} \)
7 \( 1 + (2.63 + 2.63i)T + 7iT^{2} \)
11 \( 1 + (2.30 - 2.30i)T - 11iT^{2} \)
17 \( 1 + 1.34T + 17T^{2} \)
19 \( 1 + (3.58 - 3.58i)T - 19iT^{2} \)
23 \( 1 - 3.65T + 23T^{2} \)
29 \( 1 - 3.65iT - 29T^{2} \)
31 \( 1 + (-0.321 + 0.321i)T - 31iT^{2} \)
37 \( 1 + (-1.95 - 1.95i)T + 37iT^{2} \)
41 \( 1 + (7.89 + 7.89i)T + 41iT^{2} \)
43 \( 1 + 9.10iT - 43T^{2} \)
47 \( 1 + (-4.72 + 4.72i)T - 47iT^{2} \)
53 \( 1 - 0.216iT - 53T^{2} \)
59 \( 1 + (3.65 - 3.65i)T - 59iT^{2} \)
61 \( 1 - 6.52T + 61T^{2} \)
67 \( 1 + (-2.26 + 2.26i)T - 67iT^{2} \)
71 \( 1 + (-0.108 - 0.108i)T + 71iT^{2} \)
73 \( 1 + (3.58 + 3.58i)T + 73iT^{2} \)
79 \( 1 - 4.09T + 79T^{2} \)
83 \( 1 + (-10.5 - 10.5i)T + 83iT^{2} \)
89 \( 1 + (-8.85 + 8.85i)T - 89iT^{2} \)
97 \( 1 + (-0.168 + 0.168i)T - 97iT^{2} \)
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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

−13.93281529555655579951616806406, −13.34024235724527360681855607753, −12.34111445518893407768745788226, −10.63034761308839526972723923362, −9.955204856640970693702424726957, −8.799004969741287636728776524747, −7.22477074679798839399526679824, −6.54950350223131483699581621226, −3.66170171373167405329985139525, −2.16294521231970995815487519377, 2.77631224388767136300601664668, 5.09243984126932801547268944305, 6.16084661545106261305615036813, 8.193887357286899242019551428131, 8.991474965099147750011110009253, 9.722223408874537875180553737954, 10.90296466376379646690816320259, 13.01713840793942274569321515193, 13.35490521620321988185556713511, 14.96541428774710529013131258705

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