Properties

Label 2-1008-21.5-c3-0-35
Degree $2$
Conductor $1008$
Sign $0.113 + 0.993i$
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  + (−10.9 + 18.9i)5-s + (12.1 − 13.9i)7-s + (45.1 − 26.0i)11-s + 54.9i·13-s + (−40.8 − 70.7i)17-s + (−113. − 65.6i)19-s + (38.0 + 21.9i)23-s + (−176. − 305. i)25-s + 238. i·29-s + (−174. + 100. i)31-s + (132. + 382. i)35-s + (12.0 − 20.8i)37-s + 102.·41-s − 119.·43-s + (20.2 − 35.1i)47-s + ⋯
L(s)  = 1  + (−0.977 + 1.69i)5-s + (0.655 − 0.755i)7-s + (1.23 − 0.714i)11-s + 1.17i·13-s + (−0.582 − 1.00i)17-s + (−1.37 − 0.792i)19-s + (0.345 + 0.199i)23-s + (−1.40 − 2.44i)25-s + 1.52i·29-s + (−1.00 + 0.582i)31-s + (0.638 + 1.84i)35-s + (0.0534 − 0.0925i)37-s + 0.389·41-s − 0.424·43-s + (0.0629 − 0.109i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.113 + 0.993i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.113 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.113 + 0.993i$
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: \((2,\ 1008,\ (\ :3/2),\ 0.113 + 0.993i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8612155715\)
\(L(\frac12)\) \(\approx\) \(0.8612155715\)
\(L(\frac{5}{2})\) 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 \( 1 + (-12.1 + 13.9i)T \)
good5 \( 1 + (10.9 - 18.9i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-45.1 + 26.0i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 54.9iT - 2.19e3T^{2} \)
17 \( 1 + (40.8 + 70.7i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (113. + 65.6i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-38.0 - 21.9i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 238. iT - 2.43e4T^{2} \)
31 \( 1 + (174. - 100. i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-12.0 + 20.8i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 102.T + 6.89e4T^{2} \)
43 \( 1 + 119.T + 7.95e4T^{2} \)
47 \( 1 + (-20.2 + 35.1i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-297. + 172. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (142. + 246. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (386. + 223. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (113. + 196. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 886. iT - 3.57e5T^{2} \)
73 \( 1 + (-6.46 + 3.73i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-404. + 700. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 943.T + 5.71e5T^{2} \)
89 \( 1 + (575. - 996. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 1.55e3iT - 9.12e5T^{2} \)
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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

−9.273453342308997680033880300556, −8.585692649019940024511729602912, −7.43101055917428511059665155140, −6.85275793714638554333891916875, −6.47524486082116192838931390898, −4.72340149843687418610702355773, −3.93793106158878578821751841846, −3.18153968685277112908997419227, −1.87261030725451226534665968439, −0.23720517336777877563950465738, 1.09617984052804353487279980484, 2.07721894755719929012539194795, 3.99205042854918964314769708409, 4.30489125757876759024586830466, 5.38224818779046335636146089735, 6.18225796502858397881713698887, 7.61298996157010019205999115362, 8.260566659590958198584382295729, 8.766919552711335515075879722953, 9.494809779226672187117200932113

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