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Properties

Label 2-588-7.4-c3-0-8
Degree $2$
Conductor $588$
Sign $0.991 - 0.126i$
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 + (2 + 3.46i)5-s + (−4.5 − 7.79i)9-s + (10 − 17.3i)11-s + 4·13-s − 12·15-s + (12 − 20.7i)17-s + (22 + 38.1i)19-s + (−36 − 62.3i)23-s + (54.5 − 94.3i)25-s + 27·27-s − 38·29-s + (92 − 159. i)31-s + (30.0 + 51.9i)33-s + (15 + 25.9i)37-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (0.178 + 0.309i)5-s + (−0.166 − 0.288i)9-s + (0.274 − 0.474i)11-s + 0.0853·13-s − 0.206·15-s + (0.171 − 0.296i)17-s + (0.265 + 0.460i)19-s + (−0.326 − 0.565i)23-s + (0.435 − 0.755i)25-s + 0.192·27-s − 0.243·29-s + (0.533 − 0.923i)31-s + (0.158 + 0.274i)33-s + (0.0666 + 0.115i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.991 - 0.126i$
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: \((2,\ 588,\ (\ :3/2),\ 0.991 - 0.126i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.808870145\)
\(L(\frac12)\) \(\approx\) \(1.808870145\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2 - 3.46i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-10 + 17.3i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 4T + 2.19e3T^{2} \)
17 \( 1 + (-12 + 20.7i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-22 - 38.1i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (36 + 62.3i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 38T + 2.43e4T^{2} \)
31 \( 1 + (-92 + 159. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-15 - 25.9i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 216T + 6.89e4T^{2} \)
43 \( 1 + 164T + 7.95e4T^{2} \)
47 \( 1 + (-260 - 450. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-73 + 126. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-230 + 398. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-314 - 543. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (278 - 481. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 592T + 3.57e5T^{2} \)
73 \( 1 + (-512 + 886. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-52 - 90.0i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 324T + 5.71e5T^{2} \)
89 \( 1 + (-448 - 775. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 920T + 9.12e5T^{2} \)
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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

−10.29756792069930784077773590280, −9.579509974861709741916292958655, −8.633058739216027614299548621656, −7.68318560042889639354538152357, −6.49030756715918419417843177882, −5.81540049302534786667771493002, −4.66798234076906158018235167844, −3.65654234049244366947305095427, −2.47159855915541263947291499495, −0.75041644127924798445666235203, 0.927249346413804176962241745199, 2.08201840542395782984467963852, 3.53110357545327346674506742778, 4.82818056158691188736877355282, 5.67987843979867536561124685037, 6.73493527671600003524256422709, 7.48580460847140619301948258256, 8.524805623920164019941017999718, 9.366696393513900621970042912594, 10.28501472986141067964717362583

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