Properties

Label 2-1008-63.59-c1-0-36
Degree $2$
Conductor $1008$
Sign $0.110 + 0.993i$
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  + (1.5 − 0.866i)3-s − 3·5-s + (2.5 − 0.866i)7-s + (1.5 − 2.59i)9-s + 1.73i·11-s + (−1.5 − 0.866i)13-s + (−4.5 + 2.59i)15-s + (1.5 − 2.59i)17-s + (4.5 − 2.59i)19-s + (3 − 3.46i)21-s − 5.19i·23-s + 4·25-s − 5.19i·27-s + (−4.5 + 2.59i)29-s + (3 − 1.73i)31-s + ⋯
L(s)  = 1  + (0.866 − 0.499i)3-s − 1.34·5-s + (0.944 − 0.327i)7-s + (0.5 − 0.866i)9-s + 0.522i·11-s + (−0.416 − 0.240i)13-s + (−1.16 + 0.670i)15-s + (0.363 − 0.630i)17-s + (1.03 − 0.596i)19-s + (0.654 − 0.755i)21-s − 1.08i·23-s + 0.800·25-s − 0.999i·27-s + (−0.835 + 0.482i)29-s + (0.538 − 0.311i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.110 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.110 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.110 + 0.993i$
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: \((2,\ 1008,\ (\ :1/2),\ 0.110 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.766710297\)
\(L(\frac12)\) \(\approx\) \(1.766710297\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 + 0.866i)T \)
7 \( 1 + (-2.5 + 0.866i)T \)
good5 \( 1 + 3T + 5T^{2} \)
11 \( 1 - 1.73iT - 11T^{2} \)
13 \( 1 + (1.5 + 0.866i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.5 + 2.59i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 5.19iT - 23T^{2} \)
29 \( 1 + (4.5 - 2.59i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3 + 1.73i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.5 + 6.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (7.5 + 4.33i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-12 - 6.92i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2 + 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.46iT - 71T^{2} \)
73 \( 1 + (4.5 + 2.59i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4 + 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.5 - 12.9i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.5 - 2.59i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.5 - 0.866i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.604973725224556736570004592838, −8.757784068156949570530425481935, −7.896357013943316189036757352021, −7.49044637993106986471876771949, −6.86518808339671049873863321449, −5.19583477218089086782252856814, −4.32202318720370620718751695799, −3.44177602631156236998996209228, −2.29935325774397855562173573792, −0.78411590172673090629700142785, 1.60219997593976960311309214279, 3.09933884996784626575563988469, 3.83009221647176563415124509143, 4.71800139539361215846507755566, 5.63359605617132938065967842633, 7.21474326828636444407408044873, 7.956753267538786062637761634045, 8.235089002227363815771396552624, 9.237693193369405064129562653205, 10.06471069768450256329849144386

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