Properties

Label 2-10e2-5.4-c3-0-1
Degree $2$
Conductor $100$
Sign $0.894 - 0.447i$
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  i·3-s + 26i·7-s + 26·9-s + 45·11-s − 44i·13-s + 117i·17-s + 91·19-s + 26·21-s + 18i·23-s − 53i·27-s − 144·29-s + 26·31-s − 45i·33-s − 214i·37-s − 44·39-s + ⋯
L(s)  = 1  − 0.192i·3-s + 1.40i·7-s + 0.962·9-s + 1.23·11-s − 0.938i·13-s + 1.66i·17-s + 1.09·19-s + 0.270·21-s + 0.163i·23-s − 0.377i·27-s − 0.922·29-s + 0.150·31-s − 0.237i·33-s − 0.950i·37-s − 0.180·39-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $0.894 - 0.447i$
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: \((2,\ 100,\ (\ :3/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.65916 + 0.391675i\)
\(L(\frac12)\) \(\approx\) \(1.65916 + 0.391675i\)
\(L(\frac{5}{2})\) 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 \( 1 + iT - 27T^{2} \)
7 \( 1 - 26iT - 343T^{2} \)
11 \( 1 - 45T + 1.33e3T^{2} \)
13 \( 1 + 44iT - 2.19e3T^{2} \)
17 \( 1 - 117iT - 4.91e3T^{2} \)
19 \( 1 - 91T + 6.85e3T^{2} \)
23 \( 1 - 18iT - 1.21e4T^{2} \)
29 \( 1 + 144T + 2.43e4T^{2} \)
31 \( 1 - 26T + 2.97e4T^{2} \)
37 \( 1 + 214iT - 5.06e4T^{2} \)
41 \( 1 + 459T + 6.89e4T^{2} \)
43 \( 1 - 460iT - 7.95e4T^{2} \)
47 \( 1 + 468iT - 1.03e5T^{2} \)
53 \( 1 + 558iT - 1.48e5T^{2} \)
59 \( 1 - 72T + 2.05e5T^{2} \)
61 \( 1 + 118T + 2.26e5T^{2} \)
67 \( 1 - 251iT - 3.00e5T^{2} \)
71 \( 1 - 108T + 3.57e5T^{2} \)
73 \( 1 + 299iT - 3.89e5T^{2} \)
79 \( 1 - 898T + 4.93e5T^{2} \)
83 \( 1 + 927iT - 5.71e5T^{2} \)
89 \( 1 + 351T + 7.04e5T^{2} \)
97 \( 1 - 386iT - 9.12e5T^{2} \)
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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

−13.23685780285049394962348753335, −12.42442482096761679933996178079, −11.57849437813299749432542718192, −10.13399282547714886175612445311, −9.099306185678696802390083690620, −7.985330032480269273994850694375, −6.53777620311412573802084483033, −5.41007124147762024486604492193, −3.63260068562254612262995119738, −1.69036601213490694776935665105, 1.20542120539560899129498037758, 3.70804106047148059952631965846, 4.74320459809182051351148795248, 6.79863395467540329750603402373, 7.38174822065435501609993085107, 9.233911830151405501417135977977, 9.949827041776075875339728072208, 11.21835019734493481088449009373, 12.12961248404188290857981544684, 13.72918646526290778297024060138

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