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Properties

Label 2-720-80.69-c1-0-25
Degree $2$
Conductor $720$
Sign $0.397 - 0.917i$
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  + (1.36 + 0.386i)2-s + (1.70 + 1.05i)4-s + (1.98 + 1.03i)5-s − 3.91·7-s + (1.90 + 2.08i)8-s + (2.30 + 2.16i)10-s + (2.93 + 2.93i)11-s + (−0.732 − 0.732i)13-s + (−5.33 − 1.51i)14-s + (1.78 + 3.57i)16-s + 2.89i·17-s + (1.67 − 1.67i)19-s + (2.29 + 3.84i)20-s + (2.85 + 5.12i)22-s + 1.73·23-s + ⋯
L(s)  = 1  + (0.961 + 0.273i)2-s + (0.850 + 0.525i)4-s + (0.887 + 0.461i)5-s − 1.48·7-s + (0.674 + 0.738i)8-s + (0.727 + 0.686i)10-s + (0.884 + 0.884i)11-s + (−0.203 − 0.203i)13-s + (−1.42 − 0.404i)14-s + (0.447 + 0.894i)16-s + 0.701i·17-s + (0.384 − 0.384i)19-s + (0.512 + 0.858i)20-s + (0.609 + 1.09i)22-s + 0.361·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.397 - 0.917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.397 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.397 - 0.917i$
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: \((2,\ 720,\ (\ :1/2),\ 0.397 - 0.917i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.39022 + 1.57028i\)
\(L(\frac12)\) \(\approx\) \(2.39022 + 1.57028i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36 - 0.386i)T \)
3 \( 1 \)
5 \( 1 + (-1.98 - 1.03i)T \)
good7 \( 1 + 3.91T + 7T^{2} \)
11 \( 1 + (-2.93 - 2.93i)T + 11iT^{2} \)
13 \( 1 + (0.732 + 0.732i)T + 13iT^{2} \)
17 \( 1 - 2.89iT - 17T^{2} \)
19 \( 1 + (-1.67 + 1.67i)T - 19iT^{2} \)
23 \( 1 - 1.73T + 23T^{2} \)
29 \( 1 + (-4.99 + 4.99i)T - 29iT^{2} \)
31 \( 1 + 10.8T + 31T^{2} \)
37 \( 1 + (-6.41 + 6.41i)T - 37iT^{2} \)
41 \( 1 - 0.00577iT - 41T^{2} \)
43 \( 1 + (2.23 - 2.23i)T - 43iT^{2} \)
47 \( 1 + 11.6iT - 47T^{2} \)
53 \( 1 + (5.55 - 5.55i)T - 53iT^{2} \)
59 \( 1 + (3.83 + 3.83i)T + 59iT^{2} \)
61 \( 1 + (-9.30 + 9.30i)T - 61iT^{2} \)
67 \( 1 + (3.85 + 3.85i)T + 67iT^{2} \)
71 \( 1 - 1.15iT - 71T^{2} \)
73 \( 1 + 7.98T + 73T^{2} \)
79 \( 1 - 0.843T + 79T^{2} \)
83 \( 1 + (-5.20 - 5.20i)T + 83iT^{2} \)
89 \( 1 + 5.40iT - 89T^{2} \)
97 \( 1 - 2.24iT - 97T^{2} \)
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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

−10.54876100937157430197482435673, −9.753284897057580545856755208441, −9.074450148076866531531496583674, −7.54240377375516532848352721992, −6.72332480068235160821199826017, −6.24639882629255763336315599386, −5.30991975935683445135177693954, −4.01789638325007549151683843701, −3.10805322283004915406037390399, −2.01633404485487163701876565365, 1.20349830336800422028693385988, 2.77343461064714414943050543010, 3.55909065699671251965450208960, 4.81813657959139278444470323587, 5.84515377604089749584674157975, 6.39014966637964342168899095234, 7.22726517209700037262069150254, 8.881586549383386363178174519834, 9.526843142670733691798331265299, 10.21306500729538652459439100839

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