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Properties

Label 2-10e2-4.3-c8-0-14
Degree $2$
Conductor $100$
Sign $-0.218 + 0.975i$
Analytic cond. $40.7378$
Root an. cond. $6.38262$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (10 + 12.4i)2-s + 99.9i·3-s + (−56 + 249. i)4-s + (−1.24e3 + 999. i)6-s + 1.39e3i·7-s + (−3.68e3 + 1.79e3i)8-s − 3.42e3·9-s + 1.84e4i·11-s + (−2.49e4 − 5.59e3i)12-s + 5.47e3·13-s + (−1.74e4 + 1.39e4i)14-s + (−5.92e4 − 2.79e4i)16-s − 7.30e4·17-s + (−3.42e4 − 4.27e4i)18-s + 1.94e4i·19-s + ⋯
L(s)  = 1  + (0.625 + 0.780i)2-s + 1.23i·3-s + (−0.218 + 0.975i)4-s + (−0.962 + 0.770i)6-s + 0.582i·7-s + (−0.898 + 0.439i)8-s − 0.521·9-s + 1.26i·11-s + (−1.20 − 0.269i)12-s + 0.191·13-s + (−0.454 + 0.364i)14-s + (−0.904 − 0.426i)16-s − 0.875·17-s + (−0.326 − 0.407i)18-s + 0.149i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.218 + 0.975i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.218 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $-0.218 + 0.975i$
Analytic conductor: \(40.7378\)
Root analytic conductor: \(6.38262\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{100} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 100,\ (\ :4),\ -0.218 + 0.975i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.22183 - 1.52606i\)
\(L(\frac12)\) \(\approx\) \(1.22183 - 1.52606i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-10 - 12.4i)T \)
5 \( 1 \)
good3 \( 1 - 99.9iT - 6.56e3T^{2} \)
7 \( 1 - 1.39e3iT - 5.76e6T^{2} \)
11 \( 1 - 1.84e4iT - 2.14e8T^{2} \)
13 \( 1 - 5.47e3T + 8.15e8T^{2} \)
17 \( 1 + 7.30e4T + 6.97e9T^{2} \)
19 \( 1 - 1.94e4iT - 1.69e10T^{2} \)
23 \( 1 + 2.37e5iT - 7.83e10T^{2} \)
29 \( 1 + 1.28e5T + 5.00e11T^{2} \)
31 \( 1 - 6.79e4iT - 8.52e11T^{2} \)
37 \( 1 - 3.47e6T + 3.51e12T^{2} \)
41 \( 1 - 2.14e6T + 7.98e12T^{2} \)
43 \( 1 + 5.92e6iT - 1.16e13T^{2} \)
47 \( 1 - 7.62e6iT - 2.38e13T^{2} \)
53 \( 1 + 8.24e5T + 6.22e13T^{2} \)
59 \( 1 - 3.72e6iT - 1.46e14T^{2} \)
61 \( 1 + 1.47e7T + 1.91e14T^{2} \)
67 \( 1 - 1.52e7iT - 4.06e14T^{2} \)
71 \( 1 - 1.19e6iT - 6.45e14T^{2} \)
73 \( 1 - 5.72e6T + 8.06e14T^{2} \)
79 \( 1 + 3.59e7iT - 1.51e15T^{2} \)
83 \( 1 + 5.19e7iT - 2.25e15T^{2} \)
89 \( 1 + 8.33e7T + 3.93e15T^{2} \)
97 \( 1 + 1.20e8T + 7.83e15T^{2} \)
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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

−13.06274448365495508289800426144, −12.16523050872682017951143637997, −10.88406627294641026297676485328, −9.597106663453854062282133004488, −8.769274987907796786415304895002, −7.35401758166011095200879046607, −6.02763873878261766002209968218, −4.76364072246339087921857912606, −4.09831564197085165340472669269, −2.52555440435739325971294378086, 0.45430277484204201231094552961, 1.39123682683298762205799344567, 2.75124450073670663198300714436, 4.14735060784961943484962439741, 5.79585884636010310450055239232, 6.74764006402317521654792801658, 8.079854756701911375413140979822, 9.457158896145852434135632415622, 10.91487486490400093165662042445, 11.54449094768855506138710161045

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