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Properties

Label 2-1008-7.4-c1-0-9
Degree $2$
Conductor $1008$
Sign $0.605 + 0.795i$
Analytic cond. $8.04892$
Root an. cond. $2.83706$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.5 − 0.866i)7-s + 5·13-s + (−0.5 − 0.866i)19-s + (2.5 − 4.33i)25-s + (5.5 − 9.52i)31-s + (−5.5 − 9.52i)37-s + 13·43-s + (5.5 + 4.33i)49-s + (−7 − 12.1i)61-s + (2.5 − 4.33i)67-s + (−8.5 + 14.7i)73-s + (8.5 + 14.7i)79-s + (−12.5 − 4.33i)91-s + 14·97-s + (−6.5 − 11.2i)103-s + ⋯
L(s)  = 1  + (−0.944 − 0.327i)7-s + 1.38·13-s + (−0.114 − 0.198i)19-s + (0.5 − 0.866i)25-s + (0.987 − 1.71i)31-s + (−0.904 − 1.56i)37-s + 1.98·43-s + (0.785 + 0.618i)49-s + (−0.896 − 1.55i)61-s + (0.305 − 0.529i)67-s + (−0.994 + 1.72i)73-s + (0.956 + 1.65i)79-s + (−1.31 − 0.453i)91-s + 1.42·97-s + (−0.640 − 1.10i)103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.605 + 0.795i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (865, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.605 + 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.393708800\)
\(L(\frac12)\) \(\approx\) \(1.393708800\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.5 + 0.866i)T \)
good5 \( 1 + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 5T + 13T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (-5.5 + 9.52i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.5 + 9.52i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 13T + 43T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7 + 12.1i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.5 + 4.33i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (8.5 - 14.7i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-8.5 - 14.7i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 14T + 97T^{2} \)
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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

−9.809009393344040431841638766542, −9.066547007754401524760710694086, −8.244313826592476516769365845393, −7.26852837103918348319673462814, −6.34820103163458864365824097373, −5.78639954121360440551997836059, −4.35358191122198076322231899693, −3.60198682429587614252906657153, −2.43417936305326061586001777444, −0.72855307877304609131911679220, 1.28665101043114623421872992620, 2.90221946789673262587701705455, 3.65284904769537353053977339820, 4.88605353687940802356602423460, 5.98856391741397329669250132988, 6.54445249687713021824767128770, 7.53941968020711715228998599006, 8.725305347389657806964743522039, 9.021989802541893962256271741775, 10.22189107753831843434033443577

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