Properties

Label 2-588-7.2-c3-0-16
Degree $2$
Conductor $588$
Sign $-0.605 + 0.795i$
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 + (4.91 − 8.50i)5-s + (−4.5 + 7.79i)9-s + (7.08 + 12.2i)11-s + 26.1·13-s − 29.4·15-s + (−39.2 − 68.0i)17-s + (36.5 − 63.3i)19-s + (48 − 83.1i)23-s + (14.2 + 24.6i)25-s + 27·27-s + 173.·29-s + (33.6 + 58.2i)31-s + (21.2 − 36.8i)33-s + (150. − 261. i)37-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (0.439 − 0.761i)5-s + (−0.166 + 0.288i)9-s + (0.194 + 0.336i)11-s + 0.557·13-s − 0.507·15-s + (−0.560 − 0.971i)17-s + (0.441 − 0.765i)19-s + (0.435 − 0.753i)23-s + (0.113 + 0.197i)25-s + 0.192·27-s + 1.10·29-s + (0.194 + 0.337i)31-s + (0.112 − 0.194i)33-s + (0.670 − 1.16i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.605 + 0.795i$
Analytic conductor: \(34.6931\)
Root analytic conductor: \(5.89008\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :3/2),\ -0.605 + 0.795i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.610531701\)
\(L(\frac12)\) \(\approx\) \(1.610531701\)
\(L(\frac{5}{2})\) 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 \( 1 + (-4.91 + 8.50i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-7.08 - 12.2i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 26.1T + 2.19e3T^{2} \)
17 \( 1 + (39.2 + 68.0i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-36.5 + 63.3i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-48 + 83.1i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 173.T + 2.43e4T^{2} \)
31 \( 1 + (-33.6 - 58.2i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-150. + 261. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 472.T + 6.89e4T^{2} \)
43 \( 1 + 463.T + 7.95e4T^{2} \)
47 \( 1 + (45.5 - 78.9i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-81.6 - 141. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (300. + 520. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-285. + 495. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-269. - 467. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 1.06e3T + 3.57e5T^{2} \)
73 \( 1 + (221. + 383. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-22.8 + 39.5i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 686.T + 5.71e5T^{2} \)
89 \( 1 + (-330. + 571. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 658.T + 9.12e5T^{2} \)
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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

−9.898688802721141716381804183332, −9.020924290856002655828733276121, −8.360367244741489754631686781695, −7.10719031894088817988546950690, −6.46607861742881526716299339979, −5.24234120158685951583743009332, −4.62057047159707113028859450741, −2.97860823145643375278612658508, −1.64741665556912293653378585974, −0.51518012410538863732201885457, 1.41679011685305680249997197603, 2.94650425930451893801797998430, 3.88070649834038355842515099179, 5.11280721387812438938278068994, 6.18697242892464661326049025264, 6.69123885743128648219891912420, 8.088349794807743316634467267022, 8.864299276462659707209233146907, 10.08997644722540324480265069236, 10.36080266653139261017259074555

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