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Properties

Label 2-1008-252.115-c1-0-42
Degree $2$
Conductor $1008$
Sign $-0.998 - 0.0557i$
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  + (−0.597 − 1.62i)3-s + (2.47 + 1.42i)5-s + (−2.59 − 0.498i)7-s + (−2.28 + 1.94i)9-s + (1.95 − 1.13i)11-s + (−5.59 + 3.22i)13-s + (0.842 − 4.87i)15-s + (−4.74 − 2.74i)17-s + (−1.59 − 2.75i)19-s + (0.744 + 4.52i)21-s + (−5.17 − 2.98i)23-s + (1.58 + 2.73i)25-s + (4.52 + 2.55i)27-s + (1.06 − 1.85i)29-s − 4.03·31-s + ⋯
L(s)  = 1  + (−0.345 − 0.938i)3-s + (1.10 + 0.638i)5-s + (−0.982 − 0.188i)7-s + (−0.761 + 0.648i)9-s + (0.590 − 0.340i)11-s + (−1.55 + 0.895i)13-s + (0.217 − 1.25i)15-s + (−1.15 − 0.664i)17-s + (−0.365 − 0.632i)19-s + (0.162 + 0.986i)21-s + (−1.07 − 0.622i)23-s + (0.316 + 0.547i)25-s + (0.871 + 0.491i)27-s + (0.198 − 0.343i)29-s − 0.724·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0557i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0557i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.998 - 0.0557i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (367, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.998 - 0.0557i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3390320123\)
\(L(\frac12)\) \(\approx\) \(0.3390320123\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.597 + 1.62i)T \)
7 \( 1 + (2.59 + 0.498i)T \)
good5 \( 1 + (-2.47 - 1.42i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.95 + 1.13i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (5.59 - 3.22i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (4.74 + 2.74i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.59 + 2.75i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.17 + 2.98i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.06 + 1.85i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 4.03T + 31T^{2} \)
37 \( 1 + (5.06 + 8.77i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-9.02 + 5.20i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.0397 - 0.0229i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 5.74T + 47T^{2} \)
53 \( 1 + (2.43 - 4.22i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 6.20T + 59T^{2} \)
61 \( 1 - 6.98iT - 61T^{2} \)
67 \( 1 + 7.88iT - 67T^{2} \)
71 \( 1 - 15.5iT - 71T^{2} \)
73 \( 1 + (-11.0 - 6.38i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + 12.0iT - 79T^{2} \)
83 \( 1 + (1.88 - 3.26i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (3.95 - 2.28i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.62 - 2.67i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.434572648265090251643855051282, −8.998993988711505147629155184257, −7.50393579365536925665806865628, −6.76390831325530651785950729246, −6.42921209786028661335638017765, −5.49972860970224823807707328512, −4.23811185498461342641414562369, −2.59073052460757869221199237078, −2.13479946359382834727934606270, −0.14316667028183989799847467355, 1.95263359260301216199603919457, 3.21489101694966839847611425690, 4.36858012884445239715463627592, 5.23311144094469965054495433242, 6.00341369959106883866097879348, 6.67622267223647200608506510350, 8.103277928727853986864030586000, 9.115140332322459978596511366016, 9.724162474610821425583749653600, 10.01198443907217403661521065571

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