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Properties

Label 2-1008-63.16-c1-0-29
Degree $2$
Conductor $1008$
Sign $0.969 - 0.246i$
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  + (1.72 − 0.143i)3-s + 2.77·5-s + (−0.855 + 2.50i)7-s + (2.95 − 0.496i)9-s + 3.43·11-s + (−0.429 − 0.743i)13-s + (4.78 − 0.398i)15-s + (−0.405 − 0.701i)17-s + (−0.750 + 1.29i)19-s + (−1.11 + 4.44i)21-s − 7.64·23-s + 2.68·25-s + (5.03 − 1.28i)27-s + (3.99 − 6.92i)29-s + (−3.60 + 6.24i)31-s + ⋯
L(s)  = 1  + (0.996 − 0.0829i)3-s + 1.23·5-s + (−0.323 + 0.946i)7-s + (0.986 − 0.165i)9-s + 1.03·11-s + (−0.119 − 0.206i)13-s + (1.23 − 0.102i)15-s + (−0.0982 − 0.170i)17-s + (−0.172 + 0.298i)19-s + (−0.243 + 0.969i)21-s − 1.59·23-s + 0.536·25-s + (0.969 − 0.246i)27-s + (0.742 − 1.28i)29-s + (−0.647 + 1.12i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 - 0.246i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.969 - 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.969 - 0.246i$
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: \((2,\ 1008,\ (\ :1/2),\ 0.969 - 0.246i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.819544756\)
\(L(\frac12)\) \(\approx\) \(2.819544756\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.72 + 0.143i)T \)
7 \( 1 + (0.855 - 2.50i)T \)
good5 \( 1 - 2.77T + 5T^{2} \)
11 \( 1 - 3.43T + 11T^{2} \)
13 \( 1 + (0.429 + 0.743i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.405 + 0.701i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.750 - 1.29i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 7.64T + 23T^{2} \)
29 \( 1 + (-3.99 + 6.92i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.60 - 6.24i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.458 + 0.793i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.67 - 2.90i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.20 - 2.08i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.307 + 0.532i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.31 - 10.9i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.734 + 1.27i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.71 + 9.90i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-8.10 + 14.0i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 14.4T + 71T^{2} \)
73 \( 1 + (4.16 + 7.22i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.37 + 2.38i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.75 + 9.97i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (5.11 - 8.85i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.82 + 6.63i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.752008572500324785695421424415, −9.235813359557352498480434065887, −8.555069860654103058242759922289, −7.61662847171765251460147727471, −6.35388269839857513235843169349, −6.02367617647483052797551659382, −4.69090880023389250107082836766, −3.51355587987734014551733540254, −2.42517624906658818711334947101, −1.67237964655278979335047252589, 1.41344978148871608032415037247, 2.38290706025642774765725437928, 3.68646965191534573906574863178, 4.37075255495113931446512649109, 5.75576696715681378312169466990, 6.68250201579507789603522664379, 7.31790018669934515690095757334, 8.480261697520317817146809512698, 9.162530652969212202136395061589, 9.996123446093060037060215711143

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