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Properties

Label 2-80-5.4-c19-0-11
Degree $2$
Conductor $80$
Sign $-0.690 - 0.723i$
Analytic cond. $183.053$
Root an. cond. $13.5297$
Motivic weight $19$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.31e4i·3-s + (3.01e6 + 3.15e6i)5-s + 2.08e8i·7-s − 7.03e8·9-s − 8.06e9·11-s + 5.42e10i·13-s + (1.36e11 − 1.30e11i)15-s + 1.27e11i·17-s − 3.24e11·19-s + 8.99e12·21-s − 4.42e12i·23-s + (−8.91e11 + 1.90e13i)25-s − 1.98e13i·27-s + 1.05e14·29-s + 1.18e14·31-s + ⋯
L(s)  = 1  − 1.26i·3-s + (0.690 + 0.723i)5-s + 1.95i·7-s − 0.605·9-s − 1.03·11-s + 1.41i·13-s + (0.916 − 0.874i)15-s + 0.261i·17-s − 0.230·19-s + 2.47·21-s − 0.512i·23-s + (−0.0467 + 0.998i)25-s − 0.500i·27-s + 1.34·29-s + 0.804·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.690 - 0.723i)\, \overline{\Lambda}(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & (-0.690 - 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $-0.690 - 0.723i$
Analytic conductor: \(183.053\)
Root analytic conductor: \(13.5297\)
Motivic weight: \(19\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :19/2),\ -0.690 - 0.723i)\)

Particular Values

\(L(10)\) \(\approx\) \(1.645715654\)
\(L(\frac12)\) \(\approx\) \(1.645715654\)
\(L(\frac{21}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-3.01e6 - 3.15e6i)T \)
good3 \( 1 + 4.31e4iT - 1.16e9T^{2} \)
7 \( 1 - 2.08e8iT - 1.13e16T^{2} \)
11 \( 1 + 8.06e9T + 6.11e19T^{2} \)
13 \( 1 - 5.42e10iT - 1.46e21T^{2} \)
17 \( 1 - 1.27e11iT - 2.39e23T^{2} \)
19 \( 1 + 3.24e11T + 1.97e24T^{2} \)
23 \( 1 + 4.42e12iT - 7.46e25T^{2} \)
29 \( 1 - 1.05e14T + 6.10e27T^{2} \)
31 \( 1 - 1.18e14T + 2.16e28T^{2} \)
37 \( 1 - 1.23e15iT - 6.24e29T^{2} \)
41 \( 1 - 2.26e15T + 4.39e30T^{2} \)
43 \( 1 - 2.39e15iT - 1.08e31T^{2} \)
47 \( 1 + 2.93e15iT - 5.88e31T^{2} \)
53 \( 1 - 9.00e14iT - 5.77e32T^{2} \)
59 \( 1 + 7.94e15T + 4.42e33T^{2} \)
61 \( 1 - 8.68e16T + 8.34e33T^{2} \)
67 \( 1 + 2.07e17iT - 4.95e34T^{2} \)
71 \( 1 + 2.68e17T + 1.49e35T^{2} \)
73 \( 1 + 4.47e17iT - 2.53e35T^{2} \)
79 \( 1 + 9.56e17T + 1.13e36T^{2} \)
83 \( 1 - 9.39e17iT - 2.90e36T^{2} \)
89 \( 1 - 1.39e18T + 1.09e37T^{2} \)
97 \( 1 + 1.03e18iT - 5.60e37T^{2} \)
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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

−11.49696903271432431848645732739, −10.07016418724939439556461152746, −8.895240923303988672337395828273, −7.961143142104608818741934678099, −6.55444574953867683609488015191, −6.18482454263443635402769327761, −4.91498529320716192367437109660, −2.71201286615935276920684399644, −2.34609932457980578068312515074, −1.40968016522609995136589599136, 0.31193067663944475390245957943, 1.07039182983091830596714275759, 2.80978678279965269835673612667, 3.99369595308004773443901697223, 4.76135037543601262906679749977, 5.65976089529236510258221747773, 7.29882797256800040397669122583, 8.343000929909010046177776601952, 9.756418820678110873674269139990, 10.30007944883966027015836012772

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