Properties

Label 2-10e2-25.14-c3-0-5
Degree $2$
Conductor $100$
Sign $0.471 + 0.881i$
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  + (−5.01 − 1.62i)3-s + (9.21 + 6.33i)5-s − 9.22i·7-s + (0.635 + 0.461i)9-s + (5.78 − 4.20i)11-s + (52.4 − 72.1i)13-s + (−35.8 − 46.7i)15-s + (93.3 − 30.3i)17-s + (−37.0 − 114. i)19-s + (−15.0 + 46.2i)21-s + (−59.1 − 81.3i)23-s + (44.6 + 116. i)25-s + (81.2 + 111. i)27-s + (−59.9 + 184. i)29-s + (17.2 + 53.1i)31-s + ⋯
L(s)  = 1  + (−0.964 − 0.313i)3-s + (0.823 + 0.566i)5-s − 0.497i·7-s + (0.0235 + 0.0171i)9-s + (0.158 − 0.115i)11-s + (1.11 − 1.54i)13-s + (−0.617 − 0.805i)15-s + (1.33 − 0.432i)17-s + (−0.447 − 1.37i)19-s + (−0.156 + 0.480i)21-s + (−0.535 − 0.737i)23-s + (0.357 + 0.933i)25-s + (0.578 + 0.796i)27-s + (−0.384 + 1.18i)29-s + (0.100 + 0.307i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.471 + 0.881i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.471 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $0.471 + 0.881i$
Analytic conductor: \(5.90019\)
Root analytic conductor: \(2.42903\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{100} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 100,\ (\ :3/2),\ 0.471 + 0.881i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.06063 - 0.635379i\)
\(L(\frac12)\) \(\approx\) \(1.06063 - 0.635379i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-9.21 - 6.33i)T \)
good3 \( 1 + (5.01 + 1.62i)T + (21.8 + 15.8i)T^{2} \)
7 \( 1 + 9.22iT - 343T^{2} \)
11 \( 1 + (-5.78 + 4.20i)T + (411. - 1.26e3i)T^{2} \)
13 \( 1 + (-52.4 + 72.1i)T + (-678. - 2.08e3i)T^{2} \)
17 \( 1 + (-93.3 + 30.3i)T + (3.97e3 - 2.88e3i)T^{2} \)
19 \( 1 + (37.0 + 114. i)T + (-5.54e3 + 4.03e3i)T^{2} \)
23 \( 1 + (59.1 + 81.3i)T + (-3.75e3 + 1.15e4i)T^{2} \)
29 \( 1 + (59.9 - 184. i)T + (-1.97e4 - 1.43e4i)T^{2} \)
31 \( 1 + (-17.2 - 53.1i)T + (-2.41e4 + 1.75e4i)T^{2} \)
37 \( 1 + (14.2 - 19.6i)T + (-1.56e4 - 4.81e4i)T^{2} \)
41 \( 1 + (85.5 + 62.1i)T + (2.12e4 + 6.55e4i)T^{2} \)
43 \( 1 + 244. iT - 7.95e4T^{2} \)
47 \( 1 + (-284. - 92.3i)T + (8.39e4 + 6.10e4i)T^{2} \)
53 \( 1 + (39.6 + 12.8i)T + (1.20e5 + 8.75e4i)T^{2} \)
59 \( 1 + (-36.3 - 26.3i)T + (6.34e4 + 1.95e5i)T^{2} \)
61 \( 1 + (571. - 415. i)T + (7.01e4 - 2.15e5i)T^{2} \)
67 \( 1 + (349. - 113. i)T + (2.43e5 - 1.76e5i)T^{2} \)
71 \( 1 + (-242. + 745. i)T + (-2.89e5 - 2.10e5i)T^{2} \)
73 \( 1 + (363. + 500. i)T + (-1.20e5 + 3.69e5i)T^{2} \)
79 \( 1 + (339. - 1.04e3i)T + (-3.98e5 - 2.89e5i)T^{2} \)
83 \( 1 + (-272. + 88.5i)T + (4.62e5 - 3.36e5i)T^{2} \)
89 \( 1 + (-422. + 306. i)T + (2.17e5 - 6.70e5i)T^{2} \)
97 \( 1 + (-712. - 231. i)T + (7.38e5 + 5.36e5i)T^{2} \)
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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

−13.18150627378405523349861353810, −12.16107354327245324120357054985, −10.82549550273254992980486453197, −10.45223317495406301734082299055, −8.870997995333165062928768791552, −7.28353979195218800841376043612, −6.18268044706073674377994929449, −5.33684874240104372431505977905, −3.16923083440869796555625519809, −0.873375841077199923005146228605, 1.65736749142310142639457747067, 4.16383617750350978582244208018, 5.69955732448184033012450561178, 6.15138499040651188076144400881, 8.191105651836069940946206000639, 9.390928596223853263515983160924, 10.34001951819343472051095720133, 11.59686766925961642033545578231, 12.26225203793145567474896311075, 13.58682424715326840518428547875

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