Properties

Label 2-10e2-5.4-c1-0-0
Degree $2$
Conductor $100$
Sign $0.447 - 0.894i$
Analytic cond. $0.798504$
Root an. cond. $0.893590$
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  + 2i·3-s + 2i·7-s − 9-s − 2i·13-s − 6i·17-s + 4·19-s − 4·21-s − 6i·23-s + 4i·27-s − 6·29-s − 4·31-s + 2i·37-s + 4·39-s + 6·41-s + 10i·43-s + ⋯
L(s)  = 1  + 1.15i·3-s + 0.755i·7-s − 0.333·9-s − 0.554i·13-s − 1.45i·17-s + 0.917·19-s − 0.872·21-s − 1.25i·23-s + 0.769i·27-s − 1.11·29-s − 0.718·31-s + 0.328i·37-s + 0.640·39-s + 0.937·41-s + 1.52i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(0.798504\)
Root analytic conductor: \(0.893590\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{100} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 100,\ (\ :1/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.865143 + 0.534688i\)
\(L(\frac12)\) \(\approx\) \(0.865143 + 0.534688i\)
\(L(\frac{3}{2})\) 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 \( 1 - 2iT - 3T^{2} \)
7 \( 1 - 2iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 + 6iT - 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 10iT - 43T^{2} \)
47 \( 1 + 6iT - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 - 2iT - 67T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + 2iT - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + 6iT - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 - 2iT - 97T^{2} \)
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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

−14.33024189415308501730553576776, −12.98781540167559015103629201578, −11.81746361245409757385621519176, −10.78630433012170936477774500537, −9.678494221603797481097103622902, −8.960199626410135445348724291747, −7.43493432744747467519834720266, −5.68482641200058823791632445172, −4.61840726780746961176124773780, −3.00312250730576770700466962261, 1.62617164730213689031770455141, 3.87154079940685422623428227102, 5.81551045176731723788667961676, 7.11279555676085094514050237941, 7.79001924966433881829146023073, 9.284859191225383023872534913133, 10.62485914808997747602449648235, 11.75083487116390041285185985182, 12.81002754851319816436699890857, 13.52643273303449432817387322879

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