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Properties

Label 2-1008-63.59-c1-0-42
Degree $2$
Conductor $1008$
Sign $0.649 + 0.760i$
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  + (1.44 − 0.956i)3-s + 2.86·5-s + (1.83 − 1.90i)7-s + (1.16 − 2.76i)9-s − 2.71i·11-s + (3.18 + 1.84i)13-s + (4.14 − 2.74i)15-s + (−3.22 + 5.58i)17-s + (−2.73 + 1.58i)19-s + (0.819 − 4.50i)21-s − 2.99i·23-s + 3.22·25-s + (−0.956 − 5.10i)27-s + (−2.48 + 1.43i)29-s + (−8.26 + 4.77i)31-s + ⋯
L(s)  = 1  + (0.833 − 0.552i)3-s + 1.28·5-s + (0.692 − 0.721i)7-s + (0.389 − 0.920i)9-s − 0.817i·11-s + (0.884 + 0.510i)13-s + (1.06 − 0.708i)15-s + (−0.781 + 1.35i)17-s + (−0.628 + 0.362i)19-s + (0.178 − 0.983i)21-s − 0.623i·23-s + 0.645·25-s + (−0.184 − 0.982i)27-s + (−0.461 + 0.266i)29-s + (−1.48 + 0.857i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.649 + 0.760i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.649 + 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.649 + 0.760i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (689, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.649 + 0.760i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.825439383\)
\(L(\frac12)\) \(\approx\) \(2.825439383\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.44 + 0.956i)T \)
7 \( 1 + (-1.83 + 1.90i)T \)
good5 \( 1 - 2.86T + 5T^{2} \)
11 \( 1 + 2.71iT - 11T^{2} \)
13 \( 1 + (-3.18 - 1.84i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.22 - 5.58i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.73 - 1.58i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 2.99iT - 23T^{2} \)
29 \( 1 + (2.48 - 1.43i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (8.26 - 4.77i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.70 + 2.95i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.794 + 1.37i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.67 - 8.10i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.65 - 9.79i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.16 + 1.24i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.33 - 7.51i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.566 + 0.327i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.86 - 6.68i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 7.86iT - 71T^{2} \)
73 \( 1 + (-11.0 - 6.39i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.59 + 4.49i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.92 + 13.7i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (3.14 + 5.45i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-13.2 + 7.62i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.743164317075028720957315051634, −8.761997134699699436673871635052, −8.467709611214497537267669022369, −7.35400231788128258190837281492, −6.37182774183233830435365896578, −5.87803748052575120510291244552, −4.38019095923931581992958280179, −3.48193830041856133663311478478, −2.04538911657083844872524421044, −1.40866531474096651074412010689, 1.90940032401807792620754371165, 2.40507170569910208051814205370, 3.79013946594321190727111950943, 5.00624296822997907599349957091, 5.50172640382418980556708056070, 6.71741140692338373738997894689, 7.73208850867349345216514269331, 8.668821910546339025204607224891, 9.317387199156928473514556120268, 9.779397380842250544118771451808

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