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Properties

Label 2-1008-63.16-c1-0-10
Degree $2$
Conductor $1008$
Sign $-0.585 - 0.810i$
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  + (0.816 + 1.52i)3-s − 1.78·5-s + (−1.90 − 1.83i)7-s + (−1.66 + 2.49i)9-s + 5.61·11-s + (3.14 + 5.43i)13-s + (−1.45 − 2.72i)15-s + (0.646 + 1.11i)17-s + (−0.559 + 0.968i)19-s + (1.25 − 4.40i)21-s − 7.61·23-s − 1.81·25-s + (−5.17 − 0.507i)27-s + (−1.57 + 2.72i)29-s + (0.501 − 0.868i)31-s + ⋯
L(s)  = 1  + (0.471 + 0.881i)3-s − 0.797·5-s + (−0.718 − 0.695i)7-s + (−0.555 + 0.831i)9-s + 1.69·11-s + (0.870 + 1.50i)13-s + (−0.376 − 0.703i)15-s + (0.156 + 0.271i)17-s + (−0.128 + 0.222i)19-s + (0.274 − 0.961i)21-s − 1.58·23-s − 0.363·25-s + (−0.995 − 0.0977i)27-s + (−0.292 + 0.506i)29-s + (0.0900 − 0.156i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.585 - 0.810i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.585 - 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.585 - 0.810i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.585 - 0.810i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.243751573\)
\(L(\frac12)\) \(\approx\) \(1.243751573\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.816 - 1.52i)T \)
7 \( 1 + (1.90 + 1.83i)T \)
good5 \( 1 + 1.78T + 5T^{2} \)
11 \( 1 - 5.61T + 11T^{2} \)
13 \( 1 + (-3.14 - 5.43i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.646 - 1.11i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.559 - 0.968i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 7.61T + 23T^{2} \)
29 \( 1 + (1.57 - 2.72i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.501 + 0.868i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.96 - 10.3i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.14 - 7.17i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.34 - 4.06i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.972 + 1.68i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.45 + 7.72i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.19 + 7.26i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.41 + 4.17i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.27 - 2.21i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.86T + 71T^{2} \)
73 \( 1 + (-5.67 - 9.83i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.72 - 11.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.60 + 2.77i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.404 - 0.700i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.10 + 1.91i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.932869048354715153413606434049, −9.596620444434263001211806662998, −8.630101924413438864140776664762, −7.990355524350178667855688550421, −6.74343078345945769919724311170, −6.23114399186795332568317695611, −4.58827139873378872778082039921, −3.79564264987978966677403248852, −3.59495976572025481763930260677, −1.67003478807938881676399208848, 0.54654457054848080383615340267, 2.08027076169162477192747706937, 3.42801241282892431269012321103, 3.87450175154847720873645194207, 5.76397630564215427048321780146, 6.21836479414337239604314234772, 7.26811441284190085558522525931, 7.988458864522704775231050318554, 8.817524720274675370863544752021, 9.349637334993528544677043720362

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