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Properties

Label 2-588-7.4-c3-0-1
Degree $2$
Conductor $588$
Sign $0.198 - 0.980i$
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 + (−5.32 − 9.22i)5-s + (−4.5 − 7.79i)9-s + (3.32 − 5.76i)11-s − 75.9·13-s − 31.9·15-s + (−52.1 + 90.3i)17-s + (42.7 + 74.0i)19-s + (34.3 + 59.4i)23-s + (5.73 − 9.93i)25-s − 27·27-s + 87.7·29-s + (31.3 − 54.3i)31-s + (−9.98 − 17.2i)33-s + (−21.1 − 36.5i)37-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (−0.476 − 0.825i)5-s + (−0.166 − 0.288i)9-s + (0.0911 − 0.157i)11-s − 1.62·13-s − 0.550·15-s + (−0.743 + 1.28i)17-s + (0.516 + 0.893i)19-s + (0.311 + 0.539i)23-s + (0.0458 − 0.0794i)25-s − 0.192·27-s + 0.562·29-s + (0.181 − 0.314i)31-s + (−0.0526 − 0.0911i)33-s + (−0.0937 − 0.162i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.198 - 0.980i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.198 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.198 - 0.980i$
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: \((2,\ 588,\ (\ :3/2),\ 0.198 - 0.980i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7485686500\)
\(L(\frac12)\) \(\approx\) \(0.7485686500\)
\(L(\frac{5}{2})\) 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 \( 1 + (5.32 + 9.22i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-3.32 + 5.76i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 75.9T + 2.19e3T^{2} \)
17 \( 1 + (52.1 - 90.3i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-42.7 - 74.0i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-34.3 - 59.4i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 87.7T + 2.43e4T^{2} \)
31 \( 1 + (-31.3 + 54.3i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (21.1 + 36.5i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 313.T + 6.89e4T^{2} \)
43 \( 1 - 306.T + 7.95e4T^{2} \)
47 \( 1 + (-107. - 186. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (262. - 454. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-180. + 312. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-400. - 693. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-20.1 + 34.8i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 298.T + 3.57e5T^{2} \)
73 \( 1 + (-258. + 447. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-611. - 1.05e3i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 1.32e3T + 5.71e5T^{2} \)
89 \( 1 + (319. + 554. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 1.42e3T + 9.12e5T^{2} \)
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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

−10.40193009701628228687684733237, −9.491193075578290481623606081952, −8.587661197716016510811159137293, −7.904088135313162768608562858554, −7.07300495255026811438715234694, −5.93231398937019004285371575141, −4.84061684377700643644289810646, −3.88011235917443280849815194487, −2.49310200568791134035536497299, −1.20746125933628632855230534138, 0.22212618729462430868666746802, 2.44508423164048481878377535809, 3.13624802024893397134579629921, 4.51938462373814816953741976149, 5.17894732085374803616042421836, 6.94285454559044278536650620163, 7.13185993931058491762510084272, 8.379740668562206477843071010504, 9.391997617827593153605600163047, 10.00330709976994263479149330192

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