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Properties

Label 2-510-255.53-c1-0-29
Degree $2$
Conductor $510$
Sign $-0.241 + 0.970i$
Analytic cond. $4.07237$
Root an. cond. $2.01801$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·2-s + (1.57 − 0.711i)3-s − 4-s + (0.396 + 2.20i)5-s + (−0.711 − 1.57i)6-s + (−1.80 − 4.35i)7-s + i·8-s + (1.98 − 2.24i)9-s + (2.20 − 0.396i)10-s + (−0.688 + 1.66i)11-s + (−1.57 + 0.711i)12-s + (3.54 − 3.54i)13-s + (−4.35 + 1.80i)14-s + (2.19 + 3.19i)15-s + 16-s + (−2.12 − 3.53i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.911 − 0.410i)3-s − 0.5·4-s + (0.177 + 0.984i)5-s + (−0.290 − 0.644i)6-s + (−0.682 − 1.64i)7-s + 0.353i·8-s + (0.662 − 0.748i)9-s + (0.695 − 0.125i)10-s + (−0.207 + 0.501i)11-s + (−0.455 + 0.205i)12-s + (0.982 − 0.982i)13-s + (−1.16 + 0.482i)14-s + (0.565 + 0.824i)15-s + 0.250·16-s + (−0.514 − 0.857i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 510 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.241 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 510 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.241 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(510\)    =    \(2 \cdot 3 \cdot 5 \cdot 17\)
Sign: $-0.241 + 0.970i$
Analytic conductor: \(4.07237\)
Root analytic conductor: \(2.01801\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{510} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 510,\ (\ :1/2),\ -0.241 + 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08698 - 1.39057i\)
\(L(\frac12)\) \(\approx\) \(1.08698 - 1.39057i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (-1.57 + 0.711i)T \)
5 \( 1 + (-0.396 - 2.20i)T \)
17 \( 1 + (2.12 + 3.53i)T \)
good7 \( 1 + (1.80 + 4.35i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (0.688 - 1.66i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + (-3.54 + 3.54i)T - 13iT^{2} \)
19 \( 1 + (-2.50 + 2.50i)T - 19iT^{2} \)
23 \( 1 + (-1.64 + 0.680i)T + (16.2 - 16.2i)T^{2} \)
29 \( 1 + (-2.16 - 5.23i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + (0.226 - 0.0939i)T + (21.9 - 21.9i)T^{2} \)
37 \( 1 + (0.215 - 0.519i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (-10.5 - 4.37i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 - 3.98T + 43T^{2} \)
47 \( 1 + (3.45 - 3.45i)T - 47iT^{2} \)
53 \( 1 + 11.3T + 53T^{2} \)
59 \( 1 + (6.94 - 6.94i)T - 59iT^{2} \)
61 \( 1 + (4.54 - 10.9i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (-9.46 + 9.46i)T - 67iT^{2} \)
71 \( 1 + (-0.247 - 0.597i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (5.67 - 13.7i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (-2.72 + 6.56i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + 0.153T + 83T^{2} \)
89 \( 1 - 2.45iT - 89T^{2} \)
97 \( 1 + (-6.26 - 2.59i)T + (68.5 + 68.5i)T^{2} \)
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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

−10.62450840389760946623154554535, −9.871915020729221559316020763051, −9.129691420242865302028354867269, −7.75393779511023771456740121439, −7.21305601976423467973050550645, −6.30724028310377584074731808854, −4.47470582379097273535422154066, −3.36670184239216378154209159126, −2.80425281433315149567928349051, −1.04253845529128544995101695822, 1.97038504260826417618524714276, 3.44055223014656738251829001227, 4.54707193962560089991148508179, 5.69874165165355919015035846831, 6.32599763652999363198948613650, 7.950199674419066588848324412370, 8.587658525775291478924151811763, 9.182413514724700152541048636113, 9.677128429420309201245652234834, 11.11999095898507773396879122874

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