Properties

Label 2-10e2-4.3-c10-0-33
Degree $2$
Conductor $100$
Sign $0.0694 + 0.997i$
Analytic cond. $63.5357$
Root an. cond. $7.97093$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (23.3 + 21.8i)2-s + 416. i·3-s + (71.0 + 1.02e3i)4-s + (−9.09e3 + 9.75e3i)6-s + 9.55e3i·7-s + (−2.06e4 + 2.54e4i)8-s − 1.14e5·9-s + 3.04e5i·11-s + (−4.25e5 + 2.96e4i)12-s − 2.93e4·13-s + (−2.08e5 + 2.23e5i)14-s + (−1.03e6 + 1.45e5i)16-s + 1.45e6·17-s + (−2.68e6 − 2.50e6i)18-s − 6.37e5i·19-s + ⋯
L(s)  = 1  + (0.731 + 0.682i)2-s + 1.71i·3-s + (0.0694 + 0.997i)4-s + (−1.16 + 1.25i)6-s + 0.568i·7-s + (−0.629 + 0.776i)8-s − 1.94·9-s + 1.89i·11-s + (−1.71 + 0.119i)12-s − 0.0791·13-s + (−0.387 + 0.415i)14-s + (−0.990 + 0.138i)16-s + 1.02·17-s + (−1.41 − 1.32i)18-s − 0.257i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0694 + 0.997i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.0694 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $0.0694 + 0.997i$
Analytic conductor: \(63.5357\)
Root analytic conductor: \(7.97093\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{100} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 100,\ (\ :5),\ 0.0694 + 0.997i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(1.98809 - 1.85453i\)
\(L(\frac12)\) \(\approx\) \(1.98809 - 1.85453i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-23.3 - 21.8i)T \)
5 \( 1 \)
good3 \( 1 - 416. iT - 5.90e4T^{2} \)
7 \( 1 - 9.55e3iT - 2.82e8T^{2} \)
11 \( 1 - 3.04e5iT - 2.59e10T^{2} \)
13 \( 1 + 2.93e4T + 1.37e11T^{2} \)
17 \( 1 - 1.45e6T + 2.01e12T^{2} \)
19 \( 1 + 6.37e5iT - 6.13e12T^{2} \)
23 \( 1 + 3.42e5iT - 4.14e13T^{2} \)
29 \( 1 - 2.17e7T + 4.20e14T^{2} \)
31 \( 1 - 2.54e7iT - 8.19e14T^{2} \)
37 \( 1 + 6.66e7T + 4.80e15T^{2} \)
41 \( 1 - 9.83e7T + 1.34e16T^{2} \)
43 \( 1 - 9.92e7iT - 2.16e16T^{2} \)
47 \( 1 + 2.47e8iT - 5.25e16T^{2} \)
53 \( 1 - 4.80e8T + 1.74e17T^{2} \)
59 \( 1 + 4.28e8iT - 5.11e17T^{2} \)
61 \( 1 - 1.43e9T + 7.13e17T^{2} \)
67 \( 1 - 1.31e8iT - 1.82e18T^{2} \)
71 \( 1 - 1.02e8iT - 3.25e18T^{2} \)
73 \( 1 - 8.16e8T + 4.29e18T^{2} \)
79 \( 1 - 1.66e9iT - 9.46e18T^{2} \)
83 \( 1 - 4.49e9iT - 1.55e19T^{2} \)
89 \( 1 - 1.63e9T + 3.11e19T^{2} \)
97 \( 1 - 8.94e9T + 7.37e19T^{2} \)
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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

−12.55980656318015713665452731169, −11.80403186110435625785972516366, −10.36554819847672954052712160754, −9.497117815021463831849954209156, −8.421604707803906613966826223545, −6.98572474133434690352943924567, −5.44101138276071951357038010018, −4.77166400894260140871208970352, −3.77127357943910496509879461459, −2.52049289254321461514690293108, 0.58330512519800998475493329102, 1.10823496681648247611756986503, 2.52210264320854925818425371492, 3.60673855483355217381426252640, 5.56723578987440166422537414023, 6.33394555467801737526523742978, 7.54460381913467358122857949016, 8.694351891305858915747379614514, 10.40677103070966564106426981139, 11.45984974983626583199301222995

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