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Properties

Label 2-39e2-169.109-c0-0-0
Degree $2$
Conductor $1521$
Sign $-0.337 + 0.941i$
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  + (0.663 − 0.748i)4-s + (−1.87 − 0.585i)7-s − i·13-s + (−0.120 − 0.992i)16-s + (−0.580 + 0.580i)19-s + (0.239 − 0.970i)25-s + (−1.68 + 1.01i)28-s + (1.03 − 1.70i)31-s + (−1.63 − 0.987i)37-s + (−0.222 + 0.902i)43-s + (2.36 + 1.63i)49-s + (−0.748 − 0.663i)52-s + (1.17 + 0.616i)61-s + (−0.822 − 0.568i)64-s + (−0.103 + 1.70i)67-s + ⋯
L(s)  = 1  + (0.663 − 0.748i)4-s + (−1.87 − 0.585i)7-s − i·13-s + (−0.120 − 0.992i)16-s + (−0.580 + 0.580i)19-s + (0.239 − 0.970i)25-s + (−1.68 + 1.01i)28-s + (1.03 − 1.70i)31-s + (−1.63 − 0.987i)37-s + (−0.222 + 0.902i)43-s + (2.36 + 1.63i)49-s + (−0.748 − 0.663i)52-s + (1.17 + 0.616i)61-s + (−0.822 − 0.568i)64-s + (−0.103 + 1.70i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.337 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.337 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1521\)    =    \(3^{2} \cdot 13^{2}\)
Sign: $-0.337 + 0.941i$
Analytic conductor: \(0.759077\)
Root analytic conductor: \(0.871250\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1521} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1521,\ (\ :0),\ -0.337 + 0.941i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8736462759\)
\(L(\frac12)\) \(\approx\) \(0.8736462759\)
\(L(1)\) 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 \( 1 + (-0.663 + 0.748i)T^{2} \)
5 \( 1 + (-0.239 + 0.970i)T^{2} \)
7 \( 1 + (1.87 + 0.585i)T + (0.822 + 0.568i)T^{2} \)
11 \( 1 + (-0.663 - 0.748i)T^{2} \)
17 \( 1 + (-0.568 + 0.822i)T^{2} \)
19 \( 1 + (0.580 - 0.580i)T - iT^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (-0.748 - 0.663i)T^{2} \)
31 \( 1 + (-1.03 + 1.70i)T + (-0.464 - 0.885i)T^{2} \)
37 \( 1 + (1.63 + 0.987i)T + (0.464 + 0.885i)T^{2} \)
41 \( 1 + (0.935 + 0.354i)T^{2} \)
43 \( 1 + (0.222 - 0.902i)T + (-0.885 - 0.464i)T^{2} \)
47 \( 1 + (0.992 + 0.120i)T^{2} \)
53 \( 1 + (0.568 - 0.822i)T^{2} \)
59 \( 1 + (0.239 - 0.970i)T^{2} \)
61 \( 1 + (-1.17 - 0.616i)T + (0.568 + 0.822i)T^{2} \)
67 \( 1 + (0.103 - 1.70i)T + (-0.992 - 0.120i)T^{2} \)
71 \( 1 + (0.935 + 0.354i)T^{2} \)
73 \( 1 + (-0.506 + 1.12i)T + (-0.663 - 0.748i)T^{2} \)
79 \( 1 + (1.23 - 1.09i)T + (0.120 - 0.992i)T^{2} \)
83 \( 1 + (-0.935 + 0.354i)T^{2} \)
89 \( 1 + iT^{2} \)
97 \( 1 + (-1.11 - 0.872i)T + (0.239 + 0.970i)T^{2} \)
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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

−9.814956844494819830558244422936, −8.705792400869729513515690430986, −7.62902000639562868483909805108, −6.83580230950282890475845285858, −6.18338653884448252556103819397, −5.62635661891930645848887320830, −4.24566081283462435591223916311, −3.24522645784739387021022072595, −2.37528852830001712275706915510, −0.65172157492750589758406210152, 2.03038271139574117908248015710, 3.09412967404271745425985989543, 3.59402115319411501610425705870, 4.91658987928780581438249212026, 6.15921858958133127793020295557, 6.76302279477239156574231422675, 7.14531640280059644356503041994, 8.604320075143994613030615040049, 8.904697202589515194112392294189, 9.890359697713986842818048834186

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