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Properties

Label 2-588-28.27-c1-0-8
Degree $2$
Conductor $588$
Sign $-0.679 - 0.733i$
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  + (−0.0777 + 1.41i)2-s + 3-s + (−1.98 − 0.219i)4-s + 0.438i·5-s + (−0.0777 + 1.41i)6-s + (0.464 − 2.79i)8-s + 9-s + (−0.619 − 0.0341i)10-s + 2.11i·11-s + (−1.98 − 0.219i)12-s + 3.84i·13-s + 0.438i·15-s + (3.90 + 0.872i)16-s + 5.64i·17-s + (−0.0777 + 1.41i)18-s − 2.97·19-s + ⋯
L(s)  = 1  + (−0.0549 + 0.998i)2-s + 0.577·3-s + (−0.993 − 0.109i)4-s + 0.196i·5-s + (−0.0317 + 0.576i)6-s + (0.164 − 0.986i)8-s + 0.333·9-s + (−0.196 − 0.0107i)10-s + 0.637i·11-s + (−0.573 − 0.0633i)12-s + 1.06i·13-s + 0.113i·15-s + (0.975 + 0.218i)16-s + 1.36i·17-s + (−0.0183 + 0.332i)18-s − 0.682·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.679 - 0.733i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.679 - 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.679 - 0.733i$
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: \((2,\ 588,\ (\ :1/2),\ -0.679 - 0.733i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.569965 + 1.30479i\)
\(L(\frac12)\) \(\approx\) \(0.569965 + 1.30479i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0777 - 1.41i)T \)
3 \( 1 - T \)
7 \( 1 \)
good5 \( 1 - 0.438iT - 5T^{2} \)
11 \( 1 - 2.11iT - 11T^{2} \)
13 \( 1 - 3.84iT - 13T^{2} \)
17 \( 1 - 5.64iT - 17T^{2} \)
19 \( 1 + 2.97T + 19T^{2} \)
23 \( 1 - 4.77iT - 23T^{2} \)
29 \( 1 - 7.02T + 29T^{2} \)
31 \( 1 + 7.42T + 31T^{2} \)
37 \( 1 + 5.28T + 37T^{2} \)
41 \( 1 + 6.81iT - 41T^{2} \)
43 \( 1 + 4.38iT - 43T^{2} \)
47 \( 1 + 1.68T + 47T^{2} \)
53 \( 1 - 10.7T + 53T^{2} \)
59 \( 1 - 8.11T + 59T^{2} \)
61 \( 1 - 6.18iT - 61T^{2} \)
67 \( 1 + 7.85iT - 67T^{2} \)
71 \( 1 - 1.16iT - 71T^{2} \)
73 \( 1 - 10.0iT - 73T^{2} \)
79 \( 1 + 15.5iT - 79T^{2} \)
83 \( 1 + 5.49T + 83T^{2} \)
89 \( 1 + 10.4iT - 89T^{2} \)
97 \( 1 + 2.22iT - 97T^{2} \)
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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

−10.68428865088846048446051588195, −9.973643403666140310294353899160, −8.911733276238998215269907432491, −8.505884327604570470209676599130, −7.27456091345170233478214468976, −6.79401447219715710872125994228, −5.65326064251117149320679246765, −4.46073526954832380222010472551, −3.63863413530997206586353180598, −1.83576816117743312038232443218, 0.807695450707891344211926856982, 2.51781690064717477487020973278, 3.28816820824642996789628489241, 4.54157977742386883407900761809, 5.43583673048586046769416567906, 6.90234226968717974283787661516, 8.186467827197193146181639234964, 8.641888026823000340040455969603, 9.572555912873639634632158223812, 10.43565714685186325618607562419

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