Properties

Label 2-10e2-5.4-c3-0-0
Degree $2$
Conductor $100$
Sign $-0.447 - 0.894i$
Analytic cond. $5.90019$
Root an. cond. $2.42903$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4i·3-s + 16i·7-s + 11·9-s − 60·11-s + 86i·13-s − 18i·17-s − 44·19-s − 64·21-s + 48i·23-s + 152i·27-s + 186·29-s + 176·31-s − 240i·33-s − 254i·37-s − 344·39-s + ⋯
L(s)  = 1  + 0.769i·3-s + 0.863i·7-s + 0.407·9-s − 1.64·11-s + 1.83i·13-s − 0.256i·17-s − 0.531·19-s − 0.665·21-s + 0.435i·23-s + 1.08i·27-s + 1.19·29-s + 1.01·31-s − 1.26i·33-s − 1.12i·37-s − 1.41·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(5.90019\)
Root analytic conductor: \(2.42903\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{100} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 100,\ (\ :3/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.674470 + 1.09131i\)
\(L(\frac12)\) \(\approx\) \(0.674470 + 1.09131i\)
\(L(\frac{5}{2})\) 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 \( 1 - 4iT - 27T^{2} \)
7 \( 1 - 16iT - 343T^{2} \)
11 \( 1 + 60T + 1.33e3T^{2} \)
13 \( 1 - 86iT - 2.19e3T^{2} \)
17 \( 1 + 18iT - 4.91e3T^{2} \)
19 \( 1 + 44T + 6.85e3T^{2} \)
23 \( 1 - 48iT - 1.21e4T^{2} \)
29 \( 1 - 186T + 2.43e4T^{2} \)
31 \( 1 - 176T + 2.97e4T^{2} \)
37 \( 1 + 254iT - 5.06e4T^{2} \)
41 \( 1 - 186T + 6.89e4T^{2} \)
43 \( 1 + 100iT - 7.95e4T^{2} \)
47 \( 1 + 168iT - 1.03e5T^{2} \)
53 \( 1 + 498iT - 1.48e5T^{2} \)
59 \( 1 - 252T + 2.05e5T^{2} \)
61 \( 1 + 58T + 2.26e5T^{2} \)
67 \( 1 - 1.03e3iT - 3.00e5T^{2} \)
71 \( 1 - 168T + 3.57e5T^{2} \)
73 \( 1 - 506iT - 3.89e5T^{2} \)
79 \( 1 + 272T + 4.93e5T^{2} \)
83 \( 1 - 948iT - 5.71e5T^{2} \)
89 \( 1 - 1.01e3T + 7.04e5T^{2} \)
97 \( 1 - 766iT - 9.12e5T^{2} \)
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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

−13.76492553605150408944691994168, −12.64264936783253109661796366932, −11.54868313645862562245696587039, −10.42062042623944260433717825104, −9.478506394283487437644763242830, −8.427876360574542047146671837201, −6.92854312542438817913821164950, −5.37652633816202316078484930566, −4.26474664466351236765663714865, −2.39419980352675496741695973694, 0.73684304630943648114370250091, 2.79897712720710405170732166234, 4.73317541277584288923263248815, 6.24818868956177893080051739332, 7.62191911276852733035021536432, 8.139596549323974062959064948580, 10.24567351412919993094495006350, 10.58094451730751497712635795310, 12.36347796463015559510327025192, 13.05981286787324841657774721561

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