Properties

Label 2-39e2-13.12-c1-0-28
Degree $2$
Conductor $1521$
Sign $0.999 + 0.0304i$
Analytic cond. $12.1452$
Root an. cond. $3.48500$
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  − 0.445i·2-s + 1.80·4-s + 0.246i·5-s − 1.75i·7-s − 1.69i·8-s + 0.109·10-s + 5.65i·11-s − 0.780·14-s + 2.85·16-s − 3.80·17-s + 5.58i·19-s + 0.445i·20-s + 2.51·22-s + 8.34·23-s + 4.93·25-s + ⋯
L(s)  = 1  − 0.314i·2-s + 0.900·4-s + 0.110i·5-s − 0.662i·7-s − 0.598i·8-s + 0.0347·10-s + 1.70i·11-s − 0.208·14-s + 0.712·16-s − 0.922·17-s + 1.28i·19-s + 0.0995i·20-s + 0.536·22-s + 1.74·23-s + 0.987·25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0304i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0304i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1521\)    =    \(3^{2} \cdot 13^{2}\)
Sign: $0.999 + 0.0304i$
Analytic conductor: \(12.1452\)
Root analytic conductor: \(3.48500\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1521} (1351, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1521,\ (\ :1/2),\ 0.999 + 0.0304i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.207072551\)
\(L(\frac12)\) \(\approx\) \(2.207072551\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + 0.445iT - 2T^{2} \)
5 \( 1 - 0.246iT - 5T^{2} \)
7 \( 1 + 1.75iT - 7T^{2} \)
11 \( 1 - 5.65iT - 11T^{2} \)
17 \( 1 + 3.80T + 17T^{2} \)
19 \( 1 - 5.58iT - 19T^{2} \)
23 \( 1 - 8.34T + 23T^{2} \)
29 \( 1 - 5.93T + 29T^{2} \)
31 \( 1 - 5.26iT - 31T^{2} \)
37 \( 1 + 3.19iT - 37T^{2} \)
41 \( 1 + 0.445iT - 41T^{2} \)
43 \( 1 + 1.71T + 43T^{2} \)
47 \( 1 + 6.73iT - 47T^{2} \)
53 \( 1 - 1.06T + 53T^{2} \)
59 \( 1 + 13.7iT - 59T^{2} \)
61 \( 1 + 8.51T + 61T^{2} \)
67 \( 1 - 5.96iT - 67T^{2} \)
71 \( 1 - 5.71iT - 71T^{2} \)
73 \( 1 + 7.35iT - 73T^{2} \)
79 \( 1 - 4.45T + 79T^{2} \)
83 \( 1 - 10.1iT - 83T^{2} \)
89 \( 1 - 0.137iT - 89T^{2} \)
97 \( 1 + 13.6iT - 97T^{2} \)
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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

−9.690668421265834334970914295879, −8.706293473742422021475113145093, −7.63929466207289476574196420897, −6.90717837923794160994978228313, −6.62775814183672384332598835829, −5.20219081832251818454859301890, −4.35488474677051066076137446468, −3.30924520687169110370847047571, −2.26973820425414383438607037656, −1.27193975869277575021273776440, 0.992494258358240918918761388583, 2.64356052616917319794718643806, 3.03239759502229856393569352544, 4.64062486260569533061104556448, 5.49671165010012773147587138010, 6.33208862822294755377789555931, 6.86095582939961543880537460676, 7.88045696099171752024499573259, 8.844892250592373387560143445329, 9.001570985845364020785127966421

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