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Properties

Label 2-1008-21.5-c3-0-33
Degree $2$
Conductor $1008$
Sign $0.107 + 0.994i$
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  + (1.56 − 2.70i)5-s + (−17.1 − 6.88i)7-s + (33.2 − 19.2i)11-s − 13.8i·13-s + (47.5 + 82.4i)17-s + (−9.86 − 5.69i)19-s + (23.9 + 13.8i)23-s + (57.6 + 99.7i)25-s − 44.2i·29-s + (119. − 68.7i)31-s + (−45.4 + 35.7i)35-s + (70.6 − 122. i)37-s − 337.·41-s − 417.·43-s + (145. − 251. i)47-s + ⋯
L(s)  = 1  + (0.139 − 0.242i)5-s + (−0.928 − 0.371i)7-s + (0.912 − 0.526i)11-s − 0.294i·13-s + (0.678 + 1.17i)17-s + (−0.119 − 0.0687i)19-s + (0.217 + 0.125i)23-s + (0.460 + 0.798i)25-s − 0.283i·29-s + (0.690 − 0.398i)31-s + (−0.219 + 0.172i)35-s + (0.313 − 0.543i)37-s − 1.28·41-s − 1.48·43-s + (0.450 − 0.779i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.107 + 0.994i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.107 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.107 + 0.994i$
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: \((2,\ 1008,\ (\ :3/2),\ 0.107 + 0.994i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.673650457\)
\(L(\frac12)\) \(\approx\) \(1.673650457\)
\(L(\frac{5}{2})\) 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 \( 1 + (17.1 + 6.88i)T \)
good5 \( 1 + (-1.56 + 2.70i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-33.2 + 19.2i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 13.8iT - 2.19e3T^{2} \)
17 \( 1 + (-47.5 - 82.4i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (9.86 + 5.69i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-23.9 - 13.8i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 44.2iT - 2.43e4T^{2} \)
31 \( 1 + (-119. + 68.7i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-70.6 + 122. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 337.T + 6.89e4T^{2} \)
43 \( 1 + 417.T + 7.95e4T^{2} \)
47 \( 1 + (-145. + 251. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-14.7 + 8.53i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (299. + 519. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-459. - 265. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (325. + 563. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 934. iT - 3.57e5T^{2} \)
73 \( 1 + (-787. + 454. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-397. + 688. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 314.T + 5.71e5T^{2} \)
89 \( 1 + (179. - 310. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 80.5iT - 9.12e5T^{2} \)
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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

−9.417191021267221138931494324682, −8.638595267498229929468733074869, −7.76527424420248294763722967334, −6.66591570952730341521839325711, −6.12650386038082927614876720960, −5.08771194393254614151482353015, −3.81296473484918978066094444127, −3.23997291807927105035939234700, −1.64933024222852645521514707237, −0.48372839497376609065173246638, 1.06521378450953047207374890609, 2.50144091408982051211249129800, 3.36620565845820552226092074368, 4.50771388803979975708540349563, 5.50805330955466335002366490398, 6.70546442515159934626177103250, 6.84930499439699842961383894376, 8.210940771818472815003188419451, 9.089462001243653060560406258175, 9.779238577700329537077511139345

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