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Properties

Label 2-2008-2008.115-c0-0-0
Degree $2$
Conductor $2008$
Sign $-0.506 + 0.862i$
Analytic cond. $1.00212$
Root an. cond. $1.00106$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.728 − 0.684i)2-s + (−1.27 + 0.622i)3-s + (0.0627 − 0.998i)4-s + (−0.507 + 1.32i)6-s + (−0.637 − 0.770i)8-s + (0.633 − 0.806i)9-s + (−0.225 + 0.0170i)11-s + (0.540 + 1.31i)12-s + (−0.992 − 0.125i)16-s + (0.161 − 0.596i)17-s + (−0.0900 − 1.02i)18-s + (−0.499 − 1.67i)19-s + (−0.152 + 0.166i)22-s + (1.29 + 0.589i)24-s + (0.876 + 0.481i)25-s + ⋯
L(s)  = 1  + (0.728 − 0.684i)2-s + (−1.27 + 0.622i)3-s + (0.0627 − 0.998i)4-s + (−0.507 + 1.32i)6-s + (−0.637 − 0.770i)8-s + (0.633 − 0.806i)9-s + (−0.225 + 0.0170i)11-s + (0.540 + 1.31i)12-s + (−0.992 − 0.125i)16-s + (0.161 − 0.596i)17-s + (−0.0900 − 1.02i)18-s + (−0.499 − 1.67i)19-s + (−0.152 + 0.166i)22-s + (1.29 + 0.589i)24-s + (0.876 + 0.481i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.506 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.506 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2008\)    =    \(2^{3} \cdot 251\)
Sign: $-0.506 + 0.862i$
Analytic conductor: \(1.00212\)
Root analytic conductor: \(1.00106\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2008} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2008,\ (\ :0),\ -0.506 + 0.862i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8475144321\)
\(L(\frac12)\) \(\approx\) \(0.8475144321\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.728 + 0.684i)T \)
251 \( 1 + (0.745 + 0.666i)T \)
good3 \( 1 + (1.27 - 0.622i)T + (0.617 - 0.786i)T^{2} \)
5 \( 1 + (-0.876 - 0.481i)T^{2} \)
7 \( 1 + (-0.112 + 0.993i)T^{2} \)
11 \( 1 + (0.225 - 0.0170i)T + (0.988 - 0.150i)T^{2} \)
13 \( 1 + (-0.762 - 0.647i)T^{2} \)
17 \( 1 + (-0.161 + 0.596i)T + (-0.863 - 0.503i)T^{2} \)
19 \( 1 + (0.499 + 1.67i)T + (-0.837 + 0.546i)T^{2} \)
23 \( 1 + (-0.212 + 0.977i)T^{2} \)
29 \( 1 + (-0.693 - 0.720i)T^{2} \)
31 \( 1 + (0.888 + 0.459i)T^{2} \)
37 \( 1 + (0.997 + 0.0753i)T^{2} \)
41 \( 1 + (0.194 + 1.71i)T + (-0.974 + 0.224i)T^{2} \)
43 \( 1 + (1.77 + 0.917i)T + (0.577 + 0.816i)T^{2} \)
47 \( 1 + (-0.968 + 0.248i)T^{2} \)
53 \( 1 + (0.947 - 0.320i)T^{2} \)
59 \( 1 + (0.123 + 0.456i)T + (-0.863 + 0.503i)T^{2} \)
61 \( 1 + (-0.0125 - 0.999i)T^{2} \)
67 \( 1 + (-1.61 - 0.328i)T + (0.920 + 0.391i)T^{2} \)
71 \( 1 + (0.379 - 0.925i)T^{2} \)
73 \( 1 + (-0.838 + 0.149i)T + (0.938 - 0.344i)T^{2} \)
79 \( 1 + (0.470 + 0.882i)T^{2} \)
83 \( 1 + (0.263 - 0.0818i)T + (0.823 - 0.567i)T^{2} \)
89 \( 1 + (1.09 + 0.717i)T + (0.402 + 0.915i)T^{2} \)
97 \( 1 + (-0.00532 + 0.423i)T + (-0.999 - 0.0251i)T^{2} \)
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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

−9.394151254104424173342708165748, −8.587413857956700762511505338087, −6.99618586311108381972948088176, −6.61721278766212434234076799493, −5.37487826363116584383387048051, −5.17822128452138611690128453246, −4.34659380620340305809027410092, −3.35007687092918350804500847650, −2.22482466146044200665904332303, −0.56712921835420186911871048303, 1.57248891360875084532651169064, 3.03924836898034565526221497132, 4.15214069021573709094841368842, 5.02430611626183188071501329084, 5.75757413774147257872374221489, 6.38722597664579565958070578446, 6.86935703584150954456846369781, 7.994645988880637342867466358006, 8.301622558204073657427976946638, 9.655816287390688184873848795565

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