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Properties

Label 2-720-80.69-c1-0-15
Degree $2$
Conductor $720$
Sign $-0.0708 - 0.997i$
Analytic cond. $5.74922$
Root an. cond. $2.39775$
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  + (1 + i)2-s + 2i·4-s + (−1 − 2i)5-s + (−2 + 2i)8-s + (1 − 3i)10-s + (3 + 3i)11-s + (3 + 3i)13-s − 4·16-s + 4i·17-s + (−1 + i)19-s + (4 − 2i)20-s + 6i·22-s + 8·23-s + (−3 + 4i)25-s + 6i·26-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + i·4-s + (−0.447 − 0.894i)5-s + (−0.707 + 0.707i)8-s + (0.316 − 0.948i)10-s + (0.904 + 0.904i)11-s + (0.832 + 0.832i)13-s − 16-s + 0.970i·17-s + (−0.229 + 0.229i)19-s + (0.894 − 0.447i)20-s + 1.27i·22-s + 1.66·23-s + (−0.600 + 0.800i)25-s + 1.17i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0708 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0708 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-0.0708 - 0.997i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (469, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ -0.0708 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39962 + 1.50262i\)
\(L(\frac12)\) \(\approx\) \(1.39962 + 1.50262i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - i)T \)
3 \( 1 \)
5 \( 1 + (1 + 2i)T \)
good7 \( 1 + 7T^{2} \)
11 \( 1 + (-3 - 3i)T + 11iT^{2} \)
13 \( 1 + (-3 - 3i)T + 13iT^{2} \)
17 \( 1 - 4iT - 17T^{2} \)
19 \( 1 + (1 - i)T - 19iT^{2} \)
23 \( 1 - 8T + 23T^{2} \)
29 \( 1 + (3 - 3i)T - 29iT^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + (-3 + 3i)T - 37iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + (-3 + 3i)T - 43iT^{2} \)
47 \( 1 - 2iT - 47T^{2} \)
53 \( 1 + (-9 + 9i)T - 53iT^{2} \)
59 \( 1 + (9 + 9i)T + 59iT^{2} \)
61 \( 1 + (5 - 5i)T - 61iT^{2} \)
67 \( 1 + (3 + 3i)T + 67iT^{2} \)
71 \( 1 + 6iT - 71T^{2} \)
73 \( 1 - 6T + 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + (9 + 9i)T + 83iT^{2} \)
89 \( 1 - 12iT - 89T^{2} \)
97 \( 1 - 12iT - 97T^{2} \)
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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

−10.92023239810247055995245921225, −9.305858781316963035859469230036, −8.891372493050236921324771493443, −7.956046592461271924806820942653, −7.01385121640649142756975729189, −6.24776571890206853262596818762, −5.12062100164718091409456953175, −4.27164574432632001134687497610, −3.58170783695403999621274150417, −1.67844278156686471158517366421, 0.946333878157062641540079566239, 2.79129162062936904829559365960, 3.40861988712815530560269136261, 4.45473078778446469776359931836, 5.70991505358925597335335818308, 6.45278279873154311563427600862, 7.37932400712552283860942124974, 8.652850045205801632123609314167, 9.485014680690147277290111964959, 10.54762015453722569392200215276

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