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Properties

Label 2-720-9.7-c1-0-20
Degree $2$
Conductor $720$
Sign $0.00922 + 0.999i$
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  + (0.724 − 1.57i)3-s + (0.5 + 0.866i)5-s + (1.72 − 2.98i)7-s + (−1.94 − 2.28i)9-s + (1.72 − 0.158i)15-s − 2·17-s + 6.89·19-s + (−3.44 − 4.87i)21-s + (−3.72 − 6.45i)23-s + (−0.499 + 0.866i)25-s + (−5.00 + 1.41i)27-s + (0.949 − 1.64i)29-s + (0.550 + 0.953i)31-s + 3.44·35-s − 6·37-s + ⋯
L(s)  = 1  + (0.418 − 0.908i)3-s + (0.223 + 0.387i)5-s + (0.651 − 1.12i)7-s + (−0.649 − 0.760i)9-s + (0.445 − 0.0410i)15-s − 0.485·17-s + 1.58·19-s + (−0.752 − 1.06i)21-s + (−0.776 − 1.34i)23-s + (−0.0999 + 0.173i)25-s + (−0.962 + 0.272i)27-s + (0.176 − 0.305i)29-s + (0.0988 + 0.171i)31-s + 0.583·35-s − 0.986·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00922 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00922 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.00922 + 0.999i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ 0.00922 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29837 - 1.28644i\)
\(L(\frac12)\) \(\approx\) \(1.29837 - 1.28644i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.724 + 1.57i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
good7 \( 1 + (-1.72 + 2.98i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 - 6.89T + 19T^{2} \)
23 \( 1 + (3.72 + 6.45i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.949 + 1.64i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.550 - 0.953i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 6T + 37T^{2} \)
41 \( 1 + (-4.94 - 8.57i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.89 + 10.2i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.72 - 8.18i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 7.79T + 53T^{2} \)
59 \( 1 + (0.550 + 0.953i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.5 + 2.59i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.62 + 11.4i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 9.79T + 71T^{2} \)
73 \( 1 - 13.7T + 73T^{2} \)
79 \( 1 + (3.44 - 5.97i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.72 - 4.71i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 2.79T + 89T^{2} \)
97 \( 1 + (1 - 1.73i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.26079918218444508237033054458, −9.286671787870741268903462382014, −8.247626688955354888665877254979, −7.54942886089865946097834693420, −6.87981915569236881268368140640, −5.95429030302776918113362440536, −4.63879131146943997356033840944, −3.47539106898845180001855828377, −2.26823031383768460065273737564, −0.962778779675070972594813309367, 1.86822181613396461112825376592, 3.03881253096132605292838542846, 4.23081924586108941434091526201, 5.34128921405890531874754445330, 5.68136778805612990920863920969, 7.37066044645943103308851492332, 8.286359980126810791528012763339, 9.018392504470202848595215541374, 9.581452996931094808019792354208, 10.48885653339331397485250377547

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