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Properties

Label 2-588-7.4-c3-0-6
Degree $2$
Conductor $588$
Sign $0.198 - 0.980i$
Analytic cond. $34.6931$
Root an. cond. $5.89008$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.5 − 2.59i)3-s + (9.57 + 16.5i)5-s + (−4.5 − 7.79i)9-s + (−20.2 + 35.1i)11-s + 50.4·13-s + 57.4·15-s + (25.9 − 44.9i)17-s + (16.5 + 28.6i)19-s + (−31.4 − 54.4i)23-s + (−120. + 209. i)25-s − 27·27-s + 129.·29-s + (−121. + 209. i)31-s + (60.8 + 105. i)33-s + (194. + 337. i)37-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (0.855 + 1.48i)5-s + (−0.166 − 0.288i)9-s + (−0.556 + 0.963i)11-s + 1.07·13-s + 0.988·15-s + (0.370 − 0.641i)17-s + (0.200 + 0.346i)19-s + (−0.284 − 0.493i)23-s + (−0.965 + 1.67i)25-s − 0.192·27-s + 0.831·29-s + (−0.702 + 1.21i)31-s + (0.321 + 0.556i)33-s + (0.865 + 1.49i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.198 - 0.980i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.198 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.198 - 0.980i$
Analytic conductor: \(34.6931\)
Root analytic conductor: \(5.89008\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :3/2),\ 0.198 - 0.980i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.384856980\)
\(L(\frac12)\) \(\approx\) \(2.384856980\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.5 + 2.59i)T \)
7 \( 1 \)
good5 \( 1 + (-9.57 - 16.5i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (20.2 - 35.1i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 50.4T + 2.19e3T^{2} \)
17 \( 1 + (-25.9 + 44.9i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-16.5 - 28.6i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (31.4 + 54.4i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 129.T + 2.43e4T^{2} \)
31 \( 1 + (121. - 209. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-194. - 337. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 470.T + 6.89e4T^{2} \)
43 \( 1 + 125.T + 7.95e4T^{2} \)
47 \( 1 + (-193. - 334. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-305. + 529. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (113. - 196. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (362. + 628. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (522. - 905. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 169.T + 3.57e5T^{2} \)
73 \( 1 + (-190. + 330. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-580. - 1.00e3i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 808.T + 5.71e5T^{2} \)
89 \( 1 + (-159. - 277. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 1.13e3T + 9.12e5T^{2} \)
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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

−10.27622555023638849388090998212, −9.913646103171531409453047198178, −8.675257444616316373340385827153, −7.65899515119968871606421967196, −6.80404306369542980828646529373, −6.24296868540306730347318552301, −5.06533245567816565002574208184, −3.42612001599357359367640516347, −2.60029370897703359802217214865, −1.51707878765039385307494982604, 0.67920282034506201054384254369, 1.91123986964148619299479055720, 3.42360892210531872423600787396, 4.49793249645516268439876427986, 5.59293896997289568615599982652, 5.98795064470884751055504327166, 7.75642479184538374893655872189, 8.645773061102852908706630228082, 9.026341553834548652190399857582, 10.01901021369191738050465436437

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