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Properties

Label 2-588-28.27-c1-0-29
Degree $2$
Conductor $588$
Sign $0.437 + 0.899i$
Analytic cond. $4.69520$
Root an. cond. $2.16684$
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  + (1.34 − 0.424i)2-s − 3-s + (1.63 − 1.14i)4-s + 0.127i·5-s + (−1.34 + 0.424i)6-s + (1.72 − 2.24i)8-s + 9-s + (0.0542 + 0.172i)10-s − 3.99i·11-s + (−1.63 + 1.14i)12-s + 0.891i·13-s − 0.127i·15-s + (1.37 − 3.75i)16-s − 5.82i·17-s + (1.34 − 0.424i)18-s + 6.31·19-s + ⋯
L(s)  = 1  + (0.953 − 0.300i)2-s − 0.577·3-s + (0.819 − 0.572i)4-s + 0.0572i·5-s + (−0.550 + 0.173i)6-s + (0.610 − 0.792i)8-s + 0.333·9-s + (0.0171 + 0.0545i)10-s − 1.20i·11-s + (−0.473 + 0.330i)12-s + 0.247i·13-s − 0.0330i·15-s + (0.344 − 0.938i)16-s − 1.41i·17-s + (0.317 − 0.100i)18-s + 1.44·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.437 + 0.899i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.437 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.437 + 0.899i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (391, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ 0.437 + 0.899i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.89565 - 1.18630i\)
\(L(\frac12)\) \(\approx\) \(1.89565 - 1.18630i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.34 + 0.424i)T \)
3 \( 1 + T \)
7 \( 1 \)
good5 \( 1 - 0.127iT - 5T^{2} \)
11 \( 1 + 3.99iT - 11T^{2} \)
13 \( 1 - 0.891iT - 13T^{2} \)
17 \( 1 + 5.82iT - 17T^{2} \)
19 \( 1 - 6.31T + 19T^{2} \)
23 \( 1 - 6.60iT - 23T^{2} \)
29 \( 1 - 2.82T + 29T^{2} \)
31 \( 1 + 8.45T + 31T^{2} \)
37 \( 1 + 8.67T + 37T^{2} \)
41 \( 1 + 3.24iT - 41T^{2} \)
43 \( 1 - 0.881iT - 43T^{2} \)
47 \( 1 - 10.0T + 47T^{2} \)
53 \( 1 + 7.53T + 53T^{2} \)
59 \( 1 - 0.588T + 59T^{2} \)
61 \( 1 - 1.68iT - 61T^{2} \)
67 \( 1 - 6.35iT - 67T^{2} \)
71 \( 1 - 11.3iT - 71T^{2} \)
73 \( 1 - 15.5iT - 73T^{2} \)
79 \( 1 - 9.82iT - 79T^{2} \)
83 \( 1 - 1.48T + 83T^{2} \)
89 \( 1 + 0.449iT - 89T^{2} \)
97 \( 1 - 16.2iT - 97T^{2} \)
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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

−10.93873301607769492034881992451, −9.880049718526949717592178807210, −8.992344592908123181145719257817, −7.44172880460132907166278823251, −6.84831202103265455484578563324, −5.51791981105317144091003086524, −5.26382178844884527722893075015, −3.79577032082493720470850719437, −2.87716514962878024219291212169, −1.11574592182140202984748143234, 1.81073648823693383640880583128, 3.32452999634377592438213705804, 4.49662852296892914806397512756, 5.22455029589869457925645359582, 6.23897785105097254134281460693, 7.06107362815001621156741477334, 7.86504475381526171153961177908, 9.053134654149095156828257408863, 10.39335510167014183895701379354, 10.81371879084577325971345531170

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