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Properties

Label 2-3087-49.10-c0-0-1
Degree $2$
Conductor $3087$
Sign $-0.0305 + 0.999i$
Analytic cond. $1.54061$
Root an. cond. $1.24121$
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  + (−0.955 − 0.294i)4-s + (−0.678 − 0.541i)13-s + (0.826 + 0.563i)16-s + (1.35 − 0.781i)19-s + (−0.988 + 0.149i)25-s + (−1.68 − 0.974i)31-s + (0.425 − 0.131i)37-s + (1.12 − 0.541i)43-s + (0.488 + 0.716i)52-s + (−0.574 − 1.86i)61-s + (−0.623 − 0.781i)64-s + (0.900 − 1.56i)67-s + (−0.290 − 1.92i)73-s + (−1.52 + 0.347i)76-s + (0.222 + 0.385i)79-s + ⋯
L(s)  = 1  + (−0.955 − 0.294i)4-s + (−0.678 − 0.541i)13-s + (0.826 + 0.563i)16-s + (1.35 − 0.781i)19-s + (−0.988 + 0.149i)25-s + (−1.68 − 0.974i)31-s + (0.425 − 0.131i)37-s + (1.12 − 0.541i)43-s + (0.488 + 0.716i)52-s + (−0.574 − 1.86i)61-s + (−0.623 − 0.781i)64-s + (0.900 − 1.56i)67-s + (−0.290 − 1.92i)73-s + (−1.52 + 0.347i)76-s + (0.222 + 0.385i)79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3087 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0305 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3087 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0305 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3087\)    =    \(3^{2} \cdot 7^{3}\)
Sign: $-0.0305 + 0.999i$
Analytic conductor: \(1.54061\)
Root analytic conductor: \(1.24121\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3087} (1207, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3087,\ (\ :0),\ -0.0305 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7498559434\)
\(L(\frac12)\) \(\approx\) \(0.7498559434\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.955 + 0.294i)T^{2} \)
5 \( 1 + (0.988 - 0.149i)T^{2} \)
11 \( 1 + (-0.733 + 0.680i)T^{2} \)
13 \( 1 + (0.678 + 0.541i)T + (0.222 + 0.974i)T^{2} \)
17 \( 1 + (-0.0747 - 0.997i)T^{2} \)
19 \( 1 + (-1.35 + 0.781i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.0747 - 0.997i)T^{2} \)
29 \( 1 + (-0.900 - 0.433i)T^{2} \)
31 \( 1 + (1.68 + 0.974i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.425 + 0.131i)T + (0.826 - 0.563i)T^{2} \)
41 \( 1 + (-0.623 - 0.781i)T^{2} \)
43 \( 1 + (-1.12 + 0.541i)T + (0.623 - 0.781i)T^{2} \)
47 \( 1 + (-0.955 - 0.294i)T^{2} \)
53 \( 1 + (0.826 + 0.563i)T^{2} \)
59 \( 1 + (0.988 + 0.149i)T^{2} \)
61 \( 1 + (0.574 + 1.86i)T + (-0.826 + 0.563i)T^{2} \)
67 \( 1 + (-0.900 + 1.56i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.900 + 0.433i)T^{2} \)
73 \( 1 + (0.290 + 1.92i)T + (-0.955 + 0.294i)T^{2} \)
79 \( 1 + (-0.222 - 0.385i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.222 - 0.974i)T^{2} \)
89 \( 1 + (0.733 + 0.680i)T^{2} \)
97 \( 1 + 1.56iT - T^{2} \)
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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

−8.903822155783319589150669326237, −7.70450806835953842603454576944, −7.57962217336142893622331516649, −6.27559524529045935422165574504, −5.45678444934798681555070055475, −4.97623686133591937355791430512, −3.99859908572484897725127623213, −3.20379909572604625279486992025, −1.96309414752744157661307984109, −0.51126986263210811498919477005, 1.33052893230546012538279255029, 2.68600580678380337828137333889, 3.70284608184522680708975588263, 4.31130493462201775526943232887, 5.31022970107589522449458059488, 5.77474965645153937846399764257, 7.07578813179645070400148931245, 7.58433606107680972866692868857, 8.349257314388894367964150978595, 9.168780103393691232397105459229

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