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Properties

Label 2-510-255.128-c1-0-34
Degree $2$
Conductor $510$
Sign $-0.896 + 0.443i$
Analytic cond. $4.07237$
Root an. cond. $2.01801$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + (−0.292 + 1.70i)3-s + 4-s + (−2 − i)5-s + (−0.292 + 1.70i)6-s + (−4.12 − 1.70i)7-s + 8-s + (−2.82 − i)9-s + (−2 − i)10-s + (−0.878 − 2.12i)11-s + (−0.292 + 1.70i)12-s + (−4.41 + 4.41i)13-s + (−4.12 − 1.70i)14-s + (2.29 − 3.12i)15-s + 16-s + (−3 − 2.82i)17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.169 + 0.985i)3-s + 0.5·4-s + (−0.894 − 0.447i)5-s + (−0.119 + 0.696i)6-s + (−1.55 − 0.645i)7-s + 0.353·8-s + (−0.942 − 0.333i)9-s + (−0.632 − 0.316i)10-s + (−0.264 − 0.639i)11-s + (−0.0845 + 0.492i)12-s + (−1.22 + 1.22i)13-s + (−1.10 − 0.456i)14-s + (0.592 − 0.805i)15-s + 0.250·16-s + (−0.727 − 0.685i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 510 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.896 + 0.443i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 510 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.896 + 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(510\)    =    \(2 \cdot 3 \cdot 5 \cdot 17\)
Sign: $-0.896 + 0.443i$
Analytic conductor: \(4.07237\)
Root analytic conductor: \(2.01801\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{510} (383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 510,\ (\ :1/2),\ -0.896 + 0.443i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (0.292 - 1.70i)T \)
5 \( 1 + (2 + i)T \)
17 \( 1 + (3 + 2.82i)T \)
good7 \( 1 + (4.12 + 1.70i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (0.878 + 2.12i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (4.41 - 4.41i)T - 13iT^{2} \)
19 \( 1 + (-3.24 - 3.24i)T + 19iT^{2} \)
23 \( 1 + (1.29 - 3.12i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (-0.121 + 0.292i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + (0.707 + 0.292i)T + (21.9 + 21.9i)T^{2} \)
37 \( 1 + (-2.70 + 1.12i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (-1.12 + 0.464i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + 11.6iT - 43T^{2} \)
47 \( 1 + (5.82 - 5.82i)T - 47iT^{2} \)
53 \( 1 - 1.17iT - 53T^{2} \)
59 \( 1 + (6.41 + 6.41i)T + 59iT^{2} \)
61 \( 1 + (-1.05 - 2.53i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + (4.41 - 4.41i)T - 67iT^{2} \)
71 \( 1 + (2.36 - 5.70i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (9.36 - 3.87i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (5.87 + 14.1i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + 11.6iT - 83T^{2} \)
89 \( 1 - 14.1iT - 89T^{2} \)
97 \( 1 + (1.46 + 3.53i)T + (-68.5 + 68.5i)T^{2} \)
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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

−10.56736592206390303683446881561, −9.657675495238067394846107585318, −9.050271276348044475068938872220, −7.60105693219675964993297722093, −6.76580813224092679515554158653, −5.63779640953344731895821556045, −4.56069530493145601709720667707, −3.81636692595422431605018316862, −2.95826527610137422578981390101, 0, 2.58586006334876868139240967171, 3.11181509783481342671631650929, 4.71496501730671349550998979371, 5.88974289482620457259557891862, 6.71844236447244172384754658976, 7.38218333371259999262386275073, 8.281104797705620519597113098146, 9.630038238434669568205965591999, 10.58954507304017206588276002809

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