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Properties

Label 2-1008-9.4-c1-0-29
Degree $2$
Conductor $1008$
Sign $-0.449 + 0.893i$
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  + (−1.29 + 1.15i)3-s + (0.555 − 0.962i)5-s + (0.5 + 0.866i)7-s + (0.349 − 2.97i)9-s + (−0.944 − 1.63i)11-s + (0.5 − 0.866i)13-s + (0.388 + 1.88i)15-s − 5.87·17-s − 7.09·19-s + (−1.64 − 0.545i)21-s + (1.99 − 3.45i)23-s + (1.88 + 3.26i)25-s + (2.97 + 4.25i)27-s + (−0.493 − 0.855i)29-s + (−0.333 + 0.576i)31-s + ⋯
L(s)  = 1  + (−0.747 + 0.664i)3-s + (0.248 − 0.430i)5-s + (0.188 + 0.327i)7-s + (0.116 − 0.993i)9-s + (−0.284 − 0.493i)11-s + (0.138 − 0.240i)13-s + (0.100 + 0.486i)15-s − 1.42·17-s − 1.62·19-s + (−0.358 − 0.118i)21-s + (0.415 − 0.720i)23-s + (0.376 + 0.652i)25-s + (0.572 + 0.819i)27-s + (−0.0916 − 0.158i)29-s + (−0.0598 + 0.103i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.449 + 0.893i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.449 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.449 + 0.893i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.449 + 0.893i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4837298755\)
\(L(\frac12)\) \(\approx\) \(0.4837298755\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.29 - 1.15i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (-0.555 + 0.962i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.944 + 1.63i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 5.87T + 17T^{2} \)
19 \( 1 + 7.09T + 19T^{2} \)
23 \( 1 + (-1.99 + 3.45i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.493 + 0.855i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.333 - 0.576i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 1.33T + 37T^{2} \)
41 \( 1 + (-0.944 + 1.63i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.43 + 9.40i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.54 + 9.61i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 12.2T + 53T^{2} \)
59 \( 1 + (-2.38 + 4.12i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.88 + 3.26i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.04 + 3.54i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.7T + 71T^{2} \)
73 \( 1 + 3.09T + 73T^{2} \)
79 \( 1 + (-3.21 - 5.56i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.93 + 10.2i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 14.3T + 89T^{2} \)
97 \( 1 + (0.382 + 0.662i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.727301196865891550656652575601, −8.795439945269268940027072811206, −8.436069639575369457496223817290, −6.85065827159258799891142850631, −6.22871939482648724729094109721, −5.21908306912602478278134954361, −4.62160731199952458491714248749, −3.51781465126373213347651170676, −2.04913516522064541517931265466, −0.23217251909903247647741647832, 1.62167878771215242873806777183, 2.61866684690192721676212619497, 4.30130923997797251535066553645, 4.98727529227847827028519649379, 6.39211734609099916343107347584, 6.55368203662525336342104224594, 7.61232945483733032453003678746, 8.416454701522404598320322538511, 9.493502844715270540716100365584, 10.52906550947625689016200270913

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