Properties

Label 2-39e2-13.12-c1-0-12
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  − 2.04i·2-s − 2.19·4-s + 3.35i·5-s + 2.24i·7-s + 0.405i·8-s + 6.87·10-s − 4.93i·11-s + 4.60·14-s − 3.56·16-s + 0.911·17-s + 3.80i·19-s − 7.37i·20-s − 10.1·22-s + 2.02·23-s − 6.26·25-s + ⋯
L(s)  = 1  − 1.44i·2-s − 1.09·4-s + 1.50i·5-s + 0.849i·7-s + 0.143i·8-s + 2.17·10-s − 1.48i·11-s + 1.23·14-s − 0.891·16-s + 0.221·17-s + 0.872i·19-s − 1.64i·20-s − 2.15·22-s + 0.422·23-s − 1.25·25-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

\(L(1)\) \(\approx\) \(1.425184398\)
\(L(\frac12)\) \(\approx\) \(1.425184398\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.04iT - 2T^{2} \)
5 \( 1 - 3.35iT - 5T^{2} \)
7 \( 1 - 2.24iT - 7T^{2} \)
11 \( 1 + 4.93iT - 11T^{2} \)
17 \( 1 - 0.911T + 17T^{2} \)
19 \( 1 - 3.80iT - 19T^{2} \)
23 \( 1 - 2.02T + 23T^{2} \)
29 \( 1 - 3.93T + 29T^{2} \)
31 \( 1 - 8.82iT - 31T^{2} \)
37 \( 1 - 8.80iT - 37T^{2} \)
41 \( 1 - 6.93iT - 41T^{2} \)
43 \( 1 - 2.28T + 43T^{2} \)
47 \( 1 + 3.80iT - 47T^{2} \)
53 \( 1 + 0.542T + 53T^{2} \)
59 \( 1 - 4.71iT - 59T^{2} \)
61 \( 1 - 3.67T + 61T^{2} \)
67 \( 1 - 1.52iT - 67T^{2} \)
71 \( 1 - 2.37iT - 71T^{2} \)
73 \( 1 - 7.41iT - 73T^{2} \)
79 \( 1 + 3.74T + 79T^{2} \)
83 \( 1 - 2.30iT - 83T^{2} \)
89 \( 1 - 10.0iT - 89T^{2} \)
97 \( 1 - 16.1iT - 97T^{2} \)
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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

−9.827413637965662807462558523519, −8.821270877096831360108989148899, −8.159792087714794620653892370460, −6.86932569193475469486993256826, −6.25169915444579243676892540764, −5.26342811765687557994664339789, −3.85093449549676267761320583250, −3.01224409845638604410834341729, −2.71658151751720849364412184373, −1.31306878585220114449510101521, 0.60581511918470796796317907006, 2.12018199270485525594339372696, 4.12480374571145881596078109018, 4.65153713391220907124012058918, 5.31546685183189055151058152079, 6.27008395588013317370378702072, 7.37579691600618376620330490931, 7.49832060190092095959797448199, 8.585068610430857655632950962427, 9.186796832176717366894981255148

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