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Properties

Label 2-59e2-59.43-c0-0-0
Degree $2$
Conductor $3481$
Sign $0.365 - 0.930i$
Analytic cond. $1.73724$
Root an. cond. $1.31804$
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  + (0.161 + 0.986i)3-s + (0.907 + 0.419i)4-s + (−0.267 − 0.963i)5-s + (0.725 + 0.687i)7-s + (−0.267 + 0.963i)12-s + (0.907 − 0.419i)15-s + (0.647 + 0.762i)16-s + (−1.45 + 1.37i)17-s + (0.370 − 0.928i)19-s + (0.161 − 0.986i)20-s + (−0.561 + 0.827i)21-s + (0.468 + 0.883i)27-s + (0.370 + 0.928i)28-s + (−0.976 + 0.214i)29-s + (0.468 − 0.883i)35-s + ⋯
L(s)  = 1  + (0.161 + 0.986i)3-s + (0.907 + 0.419i)4-s + (−0.267 − 0.963i)5-s + (0.725 + 0.687i)7-s + (−0.267 + 0.963i)12-s + (0.907 − 0.419i)15-s + (0.647 + 0.762i)16-s + (−1.45 + 1.37i)17-s + (0.370 − 0.928i)19-s + (0.161 − 0.986i)20-s + (−0.561 + 0.827i)21-s + (0.468 + 0.883i)27-s + (0.370 + 0.928i)28-s + (−0.976 + 0.214i)29-s + (0.468 − 0.883i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3481 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.365 - 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3481 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.365 - 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3481\)    =    \(59^{2}\)
Sign: $0.365 - 0.930i$
Analytic conductor: \(1.73724\)
Root analytic conductor: \(1.31804\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3481} (1105, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3481,\ (\ :0),\ 0.365 - 0.930i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.767976068\)
\(L(\frac12)\) \(\approx\) \(1.767976068\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad59 \( 1 \)
good2 \( 1 + (-0.907 - 0.419i)T^{2} \)
3 \( 1 + (-0.161 - 0.986i)T + (-0.947 + 0.319i)T^{2} \)
5 \( 1 + (0.267 + 0.963i)T + (-0.856 + 0.515i)T^{2} \)
7 \( 1 + (-0.725 - 0.687i)T + (0.0541 + 0.998i)T^{2} \)
11 \( 1 + (0.161 + 0.986i)T^{2} \)
13 \( 1 + (-0.796 - 0.605i)T^{2} \)
17 \( 1 + (1.45 - 1.37i)T + (0.0541 - 0.998i)T^{2} \)
19 \( 1 + (-0.370 + 0.928i)T + (-0.725 - 0.687i)T^{2} \)
23 \( 1 + (-0.976 - 0.214i)T^{2} \)
29 \( 1 + (0.976 - 0.214i)T + (0.907 - 0.419i)T^{2} \)
31 \( 1 + (0.725 - 0.687i)T^{2} \)
37 \( 1 + (-0.267 + 0.963i)T^{2} \)
41 \( 1 + (-0.994 + 0.108i)T + (0.976 - 0.214i)T^{2} \)
43 \( 1 + (0.161 - 0.986i)T^{2} \)
47 \( 1 + (0.856 + 0.515i)T^{2} \)
53 \( 1 + (0.0541 - 0.998i)T + (-0.994 - 0.108i)T^{2} \)
61 \( 1 + (-0.907 - 0.419i)T^{2} \)
67 \( 1 + (-0.267 - 0.963i)T^{2} \)
71 \( 1 + (-0.535 + 1.92i)T + (-0.856 - 0.515i)T^{2} \)
73 \( 1 + (0.370 - 0.928i)T^{2} \)
79 \( 1 + (-0.161 + 0.986i)T + (-0.947 - 0.319i)T^{2} \)
83 \( 1 + (0.561 + 0.827i)T^{2} \)
89 \( 1 + (-0.907 + 0.419i)T^{2} \)
97 \( 1 + (0.370 + 0.928i)T^{2} \)
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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

−8.934264483514245658674134973190, −8.325095609531724886487425194393, −7.57786434503199466295932654659, −6.67269363381142436995715299715, −5.80912298649199108411996619203, −4.90233116415958881301397985477, −4.34292174754065963651858266179, −3.56915405655178696872502074147, −2.44190168001201087730604213643, −1.55767371670654220403296110562, 1.09846292079996794393159498991, 2.09045798597965785469215531734, 2.71079820818169534509055521215, 3.84237257845432515827588325197, 4.86080556086712854974323852075, 5.87476457470774939845869410833, 6.75507611749785187582051194994, 7.11731005468149587396880581980, 7.57121218180623126470596379185, 8.252467418666678027770458054251

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