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Properties

Label 2-80-5.2-c10-0-22
Degree $2$
Conductor $80$
Sign $0.224 + 0.974i$
Analytic cond. $50.8285$
Root an. cond. $7.12941$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−70.6 + 70.6i)3-s + (1.00e3 − 2.95e3i)5-s + (−2.61e3 − 2.61e3i)7-s + 4.90e4i·9-s − 1.56e4·11-s + (2.94e5 − 2.94e5i)13-s + (1.38e5 + 2.80e5i)15-s + (1.23e6 + 1.23e6i)17-s − 2.66e6i·19-s + 3.68e5·21-s + (−7.36e6 + 7.36e6i)23-s + (−7.74e6 − 5.94e6i)25-s + (−7.63e6 − 7.63e6i)27-s + 1.66e7i·29-s + 2.74e7·31-s + ⋯
L(s)  = 1  + (−0.290 + 0.290i)3-s + (0.321 − 0.946i)5-s + (−0.155 − 0.155i)7-s + 0.830i·9-s − 0.0974·11-s + (0.792 − 0.792i)13-s + (0.181 + 0.368i)15-s + (0.869 + 0.869i)17-s − 1.07i·19-s + 0.0903·21-s + (−1.14 + 1.14i)23-s + (−0.793 − 0.608i)25-s + (−0.532 − 0.532i)27-s + 0.812i·29-s + 0.958·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.224 + 0.974i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.224 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.224 + 0.974i$
Analytic conductor: \(50.8285\)
Root analytic conductor: \(7.12941\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :5),\ 0.224 + 0.974i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(1.27528 - 1.01513i\)
\(L(\frac12)\) \(\approx\) \(1.27528 - 1.01513i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-1.00e3 + 2.95e3i)T \)
good3 \( 1 + (70.6 - 70.6i)T - 5.90e4iT^{2} \)
7 \( 1 + (2.61e3 + 2.61e3i)T + 2.82e8iT^{2} \)
11 \( 1 + 1.56e4T + 2.59e10T^{2} \)
13 \( 1 + (-2.94e5 + 2.94e5i)T - 1.37e11iT^{2} \)
17 \( 1 + (-1.23e6 - 1.23e6i)T + 2.01e12iT^{2} \)
19 \( 1 + 2.66e6iT - 6.13e12T^{2} \)
23 \( 1 + (7.36e6 - 7.36e6i)T - 4.14e13iT^{2} \)
29 \( 1 - 1.66e7iT - 4.20e14T^{2} \)
31 \( 1 - 2.74e7T + 8.19e14T^{2} \)
37 \( 1 + (6.47e7 + 6.47e7i)T + 4.80e15iT^{2} \)
41 \( 1 - 5.65e7T + 1.34e16T^{2} \)
43 \( 1 + (-1.85e8 + 1.85e8i)T - 2.16e16iT^{2} \)
47 \( 1 + (6.69e7 + 6.69e7i)T + 5.25e16iT^{2} \)
53 \( 1 + (-1.86e8 + 1.86e8i)T - 1.74e17iT^{2} \)
59 \( 1 + 7.53e8iT - 5.11e17T^{2} \)
61 \( 1 + 3.22e8T + 7.13e17T^{2} \)
67 \( 1 + (9.46e8 + 9.46e8i)T + 1.82e18iT^{2} \)
71 \( 1 - 1.70e9T + 3.25e18T^{2} \)
73 \( 1 + (-8.54e8 + 8.54e8i)T - 4.29e18iT^{2} \)
79 \( 1 + 1.15e9iT - 9.46e18T^{2} \)
83 \( 1 + (-3.46e9 + 3.46e9i)T - 1.55e19iT^{2} \)
89 \( 1 + 6.69e9iT - 3.11e19T^{2} \)
97 \( 1 + (-1.86e9 - 1.86e9i)T + 7.37e19iT^{2} \)
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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

−12.17678587544826095083250118371, −10.84781723005187116712681727757, −9.989116534731042581830562432404, −8.697772333610838161773413542001, −7.69852183310979014424763003494, −5.89952022592847121834064617779, −5.08631616663658319090008049097, −3.69713949727893311792972996463, −1.85770291701091292386045061263, −0.50091227741630128720953415495, 1.14504564773601013598493108240, 2.67926200792948349869455472442, 3.98142651388087361952430007375, 5.93985255081847529871937576242, 6.54077274328506783928069398502, 7.88206440274658496213715209208, 9.396004263834110235693712659318, 10.33610604722479373060394156511, 11.60046477462895772459100823492, 12.33553245715038763000773058960

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