Properties

Label 2-1008-21.5-c3-0-37
Degree $2$
Conductor $1008$
Sign $0.330 + 0.943i$
Analytic cond. $59.4739$
Root an. cond. $7.71193$
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  + (1.35 − 2.35i)5-s + (8.74 − 16.3i)7-s + (8.39 − 4.84i)11-s − 67.7i·13-s + (50.1 + 86.7i)17-s + (59.6 + 34.4i)19-s + (126. + 73.3i)23-s + (58.8 + 101. i)25-s − 284. i·29-s + (−197. + 113. i)31-s + (−26.5 − 42.7i)35-s + (150. − 261. i)37-s − 232.·41-s − 173.·43-s + (191. − 331. i)47-s + ⋯
L(s)  = 1  + (0.121 − 0.210i)5-s + (0.471 − 0.881i)7-s + (0.229 − 0.132i)11-s − 1.44i·13-s + (0.714 + 1.23i)17-s + (0.720 + 0.416i)19-s + (1.15 + 0.664i)23-s + (0.470 + 0.814i)25-s − 1.82i·29-s + (−1.14 + 0.659i)31-s + (−0.128 − 0.206i)35-s + (0.669 − 1.16i)37-s − 0.885·41-s − 0.613·43-s + (0.594 − 1.02i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.330 + 0.943i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.330 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.330 + 0.943i$
Analytic conductor: \(59.4739\)
Root analytic conductor: \(7.71193\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :3/2),\ 0.330 + 0.943i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.359872447\)
\(L(\frac12)\) \(\approx\) \(2.359872447\)
\(L(\frac{5}{2})\) 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 \( 1 \)
7 \( 1 + (-8.74 + 16.3i)T \)
good5 \( 1 + (-1.35 + 2.35i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-8.39 + 4.84i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 67.7iT - 2.19e3T^{2} \)
17 \( 1 + (-50.1 - 86.7i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-59.6 - 34.4i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-126. - 73.3i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 284. iT - 2.43e4T^{2} \)
31 \( 1 + (197. - 113. i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-150. + 261. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 232.T + 6.89e4T^{2} \)
43 \( 1 + 173.T + 7.95e4T^{2} \)
47 \( 1 + (-191. + 331. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (22.2 - 12.8i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-371. - 644. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (343. + 198. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-293. - 508. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 501. iT - 3.57e5T^{2} \)
73 \( 1 + (-616. + 356. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (466. - 807. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 837.T + 5.71e5T^{2} \)
89 \( 1 + (-442. + 766. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 1.17e3iT - 9.12e5T^{2} \)
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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

−9.515278301818657276286265510127, −8.454954136293639421396653592898, −7.75223222043835636013118954410, −7.09158963410621901495923829702, −5.75534362310428411673375509697, −5.25066938010382642472333679642, −3.94234552611939349881812843314, −3.22929716053484872705144135946, −1.60091023261203695021384040288, −0.68019190500128271155821132944, 1.15282115465442332047997382510, 2.34631138559394585247157601002, 3.29031168079341514734732513961, 4.75039746267538162702056626270, 5.21456020016257042014041467836, 6.52594927272315191984945971114, 7.07016042232272729596441610092, 8.148565005438201243106405199738, 9.217501733367767871580976182298, 9.352098869820216783360765441094

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