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Properties

Label 2-39e2-169.125-c0-0-0
Degree $2$
Conductor $1521$
Sign $0.872 - 0.489i$
Analytic cond. $0.759077$
Root an. cond. $0.871250$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.464 − 0.885i)4-s + (−0.0744 + 1.23i)7-s + i·13-s + (−0.568 + 0.822i)16-s + (1.21 + 1.21i)19-s + (0.935 + 0.354i)25-s + (1.12 − 0.506i)28-s + (0.783 − 1.74i)31-s + (0.110 + 0.0495i)37-s + (−1.24 − 0.470i)43-s + (−0.517 − 0.0628i)49-s + (0.885 − 0.464i)52-s + (0.695 + 0.616i)61-s + (0.992 + 0.120i)64-s + (0.542 + 1.74i)67-s + ⋯
L(s)  = 1  + (−0.464 − 0.885i)4-s + (−0.0744 + 1.23i)7-s + i·13-s + (−0.568 + 0.822i)16-s + (1.21 + 1.21i)19-s + (0.935 + 0.354i)25-s + (1.12 − 0.506i)28-s + (0.783 − 1.74i)31-s + (0.110 + 0.0495i)37-s + (−1.24 − 0.470i)43-s + (−0.517 − 0.0628i)49-s + (0.885 − 0.464i)52-s + (0.695 + 0.616i)61-s + (0.992 + 0.120i)64-s + (0.542 + 1.74i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 - 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 - 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1521\)    =    \(3^{2} \cdot 13^{2}\)
Sign: $0.872 - 0.489i$
Analytic conductor: \(0.759077\)
Root analytic conductor: \(0.871250\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1521} (1477, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1521,\ (\ :0),\ 0.872 - 0.489i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9686857402\)
\(L(\frac12)\) \(\approx\) \(0.9686857402\)
\(L(1)\) 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 \( 1 - iT \)
good2 \( 1 + (0.464 + 0.885i)T^{2} \)
5 \( 1 + (-0.935 - 0.354i)T^{2} \)
7 \( 1 + (0.0744 - 1.23i)T + (-0.992 - 0.120i)T^{2} \)
11 \( 1 + (0.464 - 0.885i)T^{2} \)
17 \( 1 + (-0.120 + 0.992i)T^{2} \)
19 \( 1 + (-1.21 - 1.21i)T + iT^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (0.885 - 0.464i)T^{2} \)
31 \( 1 + (-0.783 + 1.74i)T + (-0.663 - 0.748i)T^{2} \)
37 \( 1 + (-0.110 - 0.0495i)T + (0.663 + 0.748i)T^{2} \)
41 \( 1 + (-0.239 - 0.970i)T^{2} \)
43 \( 1 + (1.24 + 0.470i)T + (0.748 + 0.663i)T^{2} \)
47 \( 1 + (0.822 - 0.568i)T^{2} \)
53 \( 1 + (0.120 - 0.992i)T^{2} \)
59 \( 1 + (0.935 + 0.354i)T^{2} \)
61 \( 1 + (-0.695 - 0.616i)T + (0.120 + 0.992i)T^{2} \)
67 \( 1 + (-0.542 - 1.74i)T + (-0.822 + 0.568i)T^{2} \)
71 \( 1 + (-0.239 - 0.970i)T^{2} \)
73 \( 1 + (0.308 - 0.186i)T + (0.464 - 0.885i)T^{2} \)
79 \( 1 + (1.75 + 0.922i)T + (0.568 + 0.822i)T^{2} \)
83 \( 1 + (0.239 - 0.970i)T^{2} \)
89 \( 1 - iT^{2} \)
97 \( 1 + (-1.39 + 0.254i)T + (0.935 - 0.354i)T^{2} \)
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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

−9.744592798178281251976427927212, −8.971956395391020851634578915264, −8.411770825488921208354051965331, −7.25803950017923064068916679670, −6.22820586828382342451422981263, −5.65797691493656274080509248419, −4.89154253505925072677784177693, −3.86829434100112678989892025541, −2.54486919537542115935995672017, −1.44702285634786165923308294102, 0.895060629984809675279029610318, 2.91459895671181273983593317950, 3.45857464109381330217861098441, 4.61111856471613261428628900939, 5.15683657613395748283971878947, 6.67862728546799448361787769928, 7.21412708870781260450570466968, 8.003866296929753581170182980425, 8.674706202679411651901147228913, 9.607422198525786735751089671751

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