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Properties

Label 2-10e3-8.5-c1-0-35
Degree $2$
Conductor $1000$
Sign $-0.480 - 0.877i$
Analytic cond. $7.98504$
Root an. cond. $2.82578$
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  + (−0.493 + 1.32i)2-s + 1.83i·3-s + (−1.51 − 1.30i)4-s + (−2.43 − 0.906i)6-s + 1.31·7-s + (2.48 − 1.35i)8-s − 0.368·9-s − 6.60i·11-s + (2.40 − 2.77i)12-s + 2.96i·13-s + (−0.649 + 1.74i)14-s + (0.574 + 3.95i)16-s + 2.69·17-s + (0.181 − 0.487i)18-s + 4.97i·19-s + ⋯
L(s)  = 1  + (−0.349 + 0.937i)2-s + 1.05i·3-s + (−0.756 − 0.654i)4-s + (−0.992 − 0.369i)6-s + 0.497·7-s + (0.877 − 0.480i)8-s − 0.122·9-s − 1.99i·11-s + (0.693 − 0.801i)12-s + 0.822i·13-s + (−0.173 + 0.465i)14-s + (0.143 + 0.989i)16-s + 0.653·17-s + (0.0428 − 0.115i)18-s + 1.14i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.480 - 0.877i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.480 - 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1000\)    =    \(2^{3} \cdot 5^{3}\)
Sign: $-0.480 - 0.877i$
Analytic conductor: \(7.98504\)
Root analytic conductor: \(2.82578\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1000} (501, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1000,\ (\ :1/2),\ -0.480 - 0.877i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.705323 + 1.19017i\)
\(L(\frac12)\) \(\approx\) \(0.705323 + 1.19017i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.493 - 1.32i)T \)
5 \( 1 \)
good3 \( 1 - 1.83iT - 3T^{2} \)
7 \( 1 - 1.31T + 7T^{2} \)
11 \( 1 + 6.60iT - 11T^{2} \)
13 \( 1 - 2.96iT - 13T^{2} \)
17 \( 1 - 2.69T + 17T^{2} \)
19 \( 1 - 4.97iT - 19T^{2} \)
23 \( 1 - 3.73T + 23T^{2} \)
29 \( 1 - 5.52iT - 29T^{2} \)
31 \( 1 - 8.36T + 31T^{2} \)
37 \( 1 + 7.19iT - 37T^{2} \)
41 \( 1 + 3.77T + 41T^{2} \)
43 \( 1 - 3.07iT - 43T^{2} \)
47 \( 1 - 8.77T + 47T^{2} \)
53 \( 1 + 0.0464iT - 53T^{2} \)
59 \( 1 + 1.02iT - 59T^{2} \)
61 \( 1 + 5.23iT - 61T^{2} \)
67 \( 1 - 10.8iT - 67T^{2} \)
71 \( 1 - 9.35T + 71T^{2} \)
73 \( 1 + 12.4T + 73T^{2} \)
79 \( 1 - 1.43T + 79T^{2} \)
83 \( 1 - 13.0iT - 83T^{2} \)
89 \( 1 - 3.94T + 89T^{2} \)
97 \( 1 + 1.00T + 97T^{2} \)
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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

−10.14324898119898556060600627932, −9.255130801322501229620149797990, −8.622478478863164538978721791775, −7.949933300171837058325147437683, −6.84343555896184440652310384651, −5.85489592940129073995158644527, −5.19540873253243356950715387351, −4.20918072918494792083683624615, −3.35725149956471953933720222072, −1.18830655386051693582383408751, 0.926473497978455344332329221674, 1.97533306534597165650988403871, 2.85932545375616187684892847839, 4.40873630239642598115040954888, 5.03267080122665256444714300666, 6.59483956597848944104481779477, 7.49023506503109278447462851038, 7.87929522339623236113312113785, 8.917273860921809490417133716083, 9.953572475115529677295268125314

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