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Properties

Label 2-588-7.4-c3-0-7
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 + (−3 − 5.19i)5-s + (−4.5 − 7.79i)9-s + (−18 + 31.1i)11-s + 62·13-s − 18·15-s + (−57 + 98.7i)17-s + (38 + 65.8i)19-s + (12 + 20.7i)23-s + (44.5 − 77.0i)25-s − 27·27-s + 54·29-s + (56 − 96.9i)31-s + (54 + 93.5i)33-s + (89 + 154. i)37-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (−0.268 − 0.464i)5-s + (−0.166 − 0.288i)9-s + (−0.493 + 0.854i)11-s + 1.32·13-s − 0.309·15-s + (−0.813 + 1.40i)17-s + (0.458 + 0.794i)19-s + (0.108 + 0.188i)23-s + (0.355 − 0.616i)25-s − 0.192·27-s + 0.345·29-s + (0.324 − 0.561i)31-s + (0.284 + 0.493i)33-s + (0.395 + 0.684i)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

\(L(2)\) \(\approx\) \(1.999926933\)
\(L(\frac12)\) \(\approx\) \(1.999926933\)
\(L(\frac{5}{2})\) 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 \( 1 + (3 + 5.19i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (18 - 31.1i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 62T + 2.19e3T^{2} \)
17 \( 1 + (57 - 98.7i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-38 - 65.8i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-12 - 20.7i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 54T + 2.43e4T^{2} \)
31 \( 1 + (-56 + 96.9i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-89 - 154. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 378T + 6.89e4T^{2} \)
43 \( 1 + 172T + 7.95e4T^{2} \)
47 \( 1 + (-96 - 166. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-201 + 348. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (198 - 342. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (127 + 219. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-506 + 876. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 840T + 3.57e5T^{2} \)
73 \( 1 + (445 - 770. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (40 + 69.2i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 108T + 5.71e5T^{2} \)
89 \( 1 + (-819 - 1.41e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 1.01e3T + 9.12e5T^{2} \)
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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

−10.34174629931237729699295735139, −9.311068297514870732874332506337, −8.315435924673506184705901777386, −7.929220549369215699759890307832, −6.66408121993122525124579538940, −5.89686915307238029727336224857, −4.57114510849333380922032992217, −3.64526255287665601643947235567, −2.18963247758081236365328139228, −1.04513199505327350551756372369, 0.70400936715459271855989799689, 2.64186921580646056510437268567, 3.40958740376852595099529052475, 4.58893211114418084095480274218, 5.62368610635681146210563997563, 6.72164535406600568040459253198, 7.63643308295150064124814948595, 8.723159257847745853314694919191, 9.201642206615003951774789274010, 10.43865124935574997480352694096

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