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Properties

Label 2-1008-7.5-c2-0-19
Degree $2$
Conductor $1008$
Sign $0.961 + 0.276i$
Analytic cond. $27.4660$
Root an. cond. $5.24080$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.51 + 2.60i)5-s + (−6.91 − 1.06i)7-s + (−4.56 − 7.89i)11-s + 3.43i·13-s + (4.73 − 2.73i)17-s + (9.45 + 5.45i)19-s + (9.95 − 17.2i)23-s + (1.09 + 1.89i)25-s + 36.7·29-s + (17.8 − 10.2i)31-s + (−28.4 − 22.8i)35-s + (23.5 − 40.8i)37-s + 41.5i·41-s + 73.9·43-s + (−51.2 − 29.5i)47-s + ⋯
L(s)  = 1  + (0.903 + 0.521i)5-s + (−0.988 − 0.152i)7-s + (−0.414 − 0.718i)11-s + 0.263i·13-s + (0.278 − 0.160i)17-s + (0.497 + 0.287i)19-s + (0.432 − 0.749i)23-s + (0.0438 + 0.0759i)25-s + 1.26·29-s + (0.574 − 0.331i)31-s + (−0.813 − 0.653i)35-s + (0.637 − 1.10i)37-s + 1.01i·41-s + 1.72·43-s + (−1.09 − 0.629i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 + 0.276i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.961 + 0.276i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.961 + 0.276i$
Analytic conductor: \(27.4660\)
Root analytic conductor: \(5.24080\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1),\ 0.961 + 0.276i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.968109977\)
\(L(\frac12)\) \(\approx\) \(1.968109977\)
\(L(2)\) 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 \( 1 + (6.91 + 1.06i)T \)
good5 \( 1 + (-4.51 - 2.60i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (4.56 + 7.89i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 3.43iT - 169T^{2} \)
17 \( 1 + (-4.73 + 2.73i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-9.45 - 5.45i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-9.95 + 17.2i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 - 36.7T + 841T^{2} \)
31 \( 1 + (-17.8 + 10.2i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-23.5 + 40.8i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 41.5iT - 1.68e3T^{2} \)
43 \( 1 - 73.9T + 1.84e3T^{2} \)
47 \( 1 + (51.2 + 29.5i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-31.7 - 54.9i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (44.8 - 25.9i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-21.2 - 12.2i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-9.58 - 16.6i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 2.54T + 5.04e3T^{2} \)
73 \( 1 + (-102. + 59.1i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-5.67 + 9.82i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 30.4iT - 6.88e3T^{2} \)
89 \( 1 + (139. + 80.6i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 101. iT - 9.40e3T^{2} \)
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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

−9.818073351797370034760343753824, −9.089187902760856004098405185098, −8.074851896381513023540970774117, −7.04786507913470513768433790029, −6.23145324636713909590930996476, −5.71887955095005801606321251263, −4.41478340793660132569707881305, −3.14204099923068791538544717946, −2.47385456595171588946317392923, −0.77675215911246644336049543061, 1.00210001873679269540354227623, 2.36302390947245037552698253769, 3.34640876794497213744332651878, 4.72697174071206057755060382931, 5.50125502610627761970059451456, 6.33568524450580391171127032555, 7.20488973647520392087719617001, 8.220397961653678715148519904141, 9.217104260052523712465874223285, 9.769685173870182750279179892524

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