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Properties

Label 2-10e3-25.19-c1-0-20
Degree $2$
Conductor $1000$
Sign $-0.395 + 0.918i$
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  + (0.403 + 0.555i)3-s − 3.54i·7-s + (0.781 − 2.40i)9-s + (0.0512 + 0.157i)11-s + (−4.33 − 1.40i)13-s + (−0.522 + 0.719i)17-s + (−2.83 − 2.05i)19-s + (1.97 − 1.43i)21-s + (−6.19 + 2.01i)23-s + (3.60 − 1.17i)27-s + (−5.90 + 4.29i)29-s + (−6.77 − 4.92i)31-s + (−0.0669 + 0.0921i)33-s + (5.86 + 1.90i)37-s + (−0.967 − 2.97i)39-s + ⋯
L(s)  = 1  + (0.232 + 0.320i)3-s − 1.34i·7-s + (0.260 − 0.801i)9-s + (0.0154 + 0.0475i)11-s + (−1.20 − 0.390i)13-s + (−0.126 + 0.174i)17-s + (−0.649 − 0.472i)19-s + (0.430 − 0.312i)21-s + (−1.29 + 0.419i)23-s + (0.694 − 0.225i)27-s + (−1.09 + 0.797i)29-s + (−1.21 − 0.884i)31-s + (−0.0116 + 0.0160i)33-s + (0.964 + 0.313i)37-s + (−0.154 − 0.476i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.395 + 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.395 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1000\)    =    \(2^{3} \cdot 5^{3}\)
Sign: $-0.395 + 0.918i$
Analytic conductor: \(7.98504\)
Root analytic conductor: \(2.82578\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1000} (849, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1000,\ (\ :1/2),\ -0.395 + 0.918i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.601038 - 0.913200i\)
\(L(\frac12)\) \(\approx\) \(0.601038 - 0.913200i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (-0.403 - 0.555i)T + (-0.927 + 2.85i)T^{2} \)
7 \( 1 + 3.54iT - 7T^{2} \)
11 \( 1 + (-0.0512 - 0.157i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (4.33 + 1.40i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.522 - 0.719i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (2.83 + 2.05i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (6.19 - 2.01i)T + (18.6 - 13.5i)T^{2} \)
29 \( 1 + (5.90 - 4.29i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (6.77 + 4.92i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-5.86 - 1.90i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (-2.69 + 8.28i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 3.40iT - 43T^{2} \)
47 \( 1 + (-0.0946 - 0.130i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-5.67 - 7.81i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-1.68 + 5.18i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-0.665 - 2.04i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-8.27 + 11.3i)T + (-20.7 - 63.7i)T^{2} \)
71 \( 1 + (-12.0 + 8.75i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-6.99 + 2.27i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (-5.42 + 3.94i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (5.29 - 7.28i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 + (-1.07 - 3.29i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-4.43 - 6.10i)T + (-29.9 + 92.2i)T^{2} \)
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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

−9.680590506865773438359697380263, −9.141220961372141069140863634631, −7.85187874298957526008490712449, −7.30610079355931160628000409865, −6.43856514765759767337411985440, −5.26606044161615135282645873169, −4.12901710269345140329029723252, −3.65688171515430859063791973126, −2.15266764889278324915443988512, −0.45322758946836536637576839728, 2.02353540963713315461341274095, 2.49700065415198748466805825088, 4.07749850804003538532752703604, 5.12054456836542230114951811959, 5.89767241725140790335040726900, 6.93128162822019905498640021442, 7.85667440135942628887654177301, 8.466553352197793732794109790122, 9.435459558307982747884924613281, 10.04471045210559353679306869743

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