Properties

Label 2-588-7.4-c5-0-7
Degree $2$
Conductor $588$
Sign $-0.991 + 0.126i$
Analytic cond. $94.3056$
Root an. cond. $9.71111$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.5 + 7.79i)3-s + (48.1 + 83.4i)5-s + (−40.5 − 70.1i)9-s + (89.7 − 155. i)11-s + 56.8·13-s − 867.·15-s + (708. − 1.22e3i)17-s + (−646. − 1.11e3i)19-s + (2.33e3 + 4.04e3i)23-s + (−3.08e3 + 5.34e3i)25-s + 729·27-s − 5.00e3·29-s + (−3.72e3 + 6.44e3i)31-s + (807. + 1.39e3i)33-s + (4.54e3 + 7.86e3i)37-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (0.862 + 1.49i)5-s + (−0.166 − 0.288i)9-s + (0.223 − 0.387i)11-s + 0.0933·13-s − 0.995·15-s + (0.594 − 1.02i)17-s + (−0.410 − 0.711i)19-s + (0.920 + 1.59i)23-s + (−0.986 + 1.70i)25-s + 0.192·27-s − 1.10·29-s + (−0.695 + 1.20i)31-s + (0.129 + 0.223i)33-s + (0.545 + 0.944i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.991 + 0.126i$
Analytic conductor: \(94.3056\)
Root analytic conductor: \(9.71111\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :5/2),\ -0.991 + 0.126i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.277680554\)
\(L(\frac12)\) \(\approx\) \(1.277680554\)
\(L(\frac{7}{2})\) 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 \( 1 + (4.5 - 7.79i)T \)
7 \( 1 \)
good5 \( 1 + (-48.1 - 83.4i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (-89.7 + 155. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 - 56.8T + 3.71e5T^{2} \)
17 \( 1 + (-708. + 1.22e3i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (646. + 1.11e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (-2.33e3 - 4.04e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + 5.00e3T + 2.05e7T^{2} \)
31 \( 1 + (3.72e3 - 6.44e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (-4.54e3 - 7.86e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + 6.62e3T + 1.15e8T^{2} \)
43 \( 1 + 1.29e4T + 1.47e8T^{2} \)
47 \( 1 + (2.74e3 + 4.76e3i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (1.47e4 - 2.55e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (-529. + 916. i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-4.74e3 - 8.21e3i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (665. - 1.15e3i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 - 4.98e4T + 1.80e9T^{2} \)
73 \( 1 + (3.78e4 - 6.56e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (3.11e4 + 5.40e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + 5.32e4T + 3.93e9T^{2} \)
89 \( 1 + (5.93e4 + 1.02e5i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 - 1.36e5T + 8.58e9T^{2} \)
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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

−10.32949793909115643354546882191, −9.665297464986891942377232098349, −8.915975746825865061395367766659, −7.39874382505056045539685478668, −6.78967333264415563300046972071, −5.81859782907067363437735560424, −5.04812317711258514115644674730, −3.46469222267719158340105835282, −2.88031988085381440231445763770, −1.47299364515494627764172170945, 0.27853016942015547246516929312, 1.39843049706441096924226273159, 2.11122929362203776944155870938, 3.93038723621104434216466081208, 4.97682392887469048212382873486, 5.77144421911097723586021623090, 6.53935802012355949372777958728, 7.85764674920836830207513292943, 8.576517921280064628400252398501, 9.403778260231890265905532034411

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