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Properties

Label 2-720-45.23-c1-0-9
Degree $2$
Conductor $720$
Sign $0.0881 - 0.996i$
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  + (0.985 + 1.42i)3-s + (2.06 + 0.863i)5-s + (−2.64 − 0.710i)7-s + (−1.05 + 2.80i)9-s + (0.765 − 0.441i)11-s + (1.35 − 0.362i)13-s + (0.803 + 3.78i)15-s + (3.83 + 3.83i)17-s + 4.46i·19-s + (−1.60 − 4.47i)21-s + (1.01 + 3.79i)23-s + (3.50 + 3.56i)25-s + (−5.04 + 1.26i)27-s + (−0.874 − 1.51i)29-s + (−2.74 + 4.74i)31-s + ⋯
L(s)  = 1  + (0.569 + 0.822i)3-s + (0.922 + 0.386i)5-s + (−1.00 − 0.268i)7-s + (−0.352 + 0.935i)9-s + (0.230 − 0.133i)11-s + (0.375 − 0.100i)13-s + (0.207 + 0.978i)15-s + (0.928 + 0.928i)17-s + 1.02i·19-s + (−0.349 − 0.976i)21-s + (0.211 + 0.790i)23-s + (0.701 + 0.712i)25-s + (−0.970 + 0.243i)27-s + (−0.162 − 0.281i)29-s + (−0.492 + 0.852i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0881 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0881 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.0881 - 0.996i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ 0.0881 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41441 + 1.29470i\)
\(L(\frac12)\) \(\approx\) \(1.41441 + 1.29470i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.985 - 1.42i)T \)
5 \( 1 + (-2.06 - 0.863i)T \)
good7 \( 1 + (2.64 + 0.710i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (-0.765 + 0.441i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.35 + 0.362i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (-3.83 - 3.83i)T + 17iT^{2} \)
19 \( 1 - 4.46iT - 19T^{2} \)
23 \( 1 + (-1.01 - 3.79i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (0.874 + 1.51i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.74 - 4.74i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.41 + 4.41i)T - 37iT^{2} \)
41 \( 1 + (5.85 + 3.37i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.28 + 4.80i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (-2.90 + 10.8i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (6.79 - 6.79i)T - 53iT^{2} \)
59 \( 1 + (-4.02 + 6.97i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.25 + 7.37i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.87 - 14.4i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 11.1iT - 71T^{2} \)
73 \( 1 + (-10.1 - 10.1i)T + 73iT^{2} \)
79 \( 1 + (-8.28 + 4.78i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.49 + 1.20i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + 11.3T + 89T^{2} \)
97 \( 1 + (-11.0 - 2.96i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.31229914898856502049015890059, −9.866497645689517234408191378188, −9.121662737282198925334978202610, −8.196681974194650704860614159803, −7.09679311973388218034470386490, −6.01360373698827807790589315574, −5.37399595552133234507048822408, −3.75250462722378424060083732772, −3.30405324459804384130872360762, −1.84890486439956094024392304878, 0.987398302801873834956204346619, 2.43308350957828553577126917099, 3.24971647495871628336809045926, 4.82736915593520387247454539096, 6.06012870808417697063672404164, 6.55764850693255474118339403922, 7.56169236071206733785656813633, 8.627452345352580881656214204510, 9.429650011668172357079277974542, 9.745226351893632840215660218531

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