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Properties

Label 2-510-255.8-c1-0-34
Degree $2$
Conductor $510$
Sign $-0.999 - 0.0153i$
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 + (−1.70 + 0.292i)3-s + 4-s + (−2 − i)5-s + (−1.70 + 0.292i)6-s + (0.121 − 0.292i)7-s + 8-s + (2.82 − i)9-s + (−2 − i)10-s + (−5.12 + 2.12i)11-s + (−1.70 + 0.292i)12-s + (−1.58 + 1.58i)13-s + (0.121 − 0.292i)14-s + (3.70 + 1.12i)15-s + 16-s + (−3 + 2.82i)17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.985 + 0.169i)3-s + 0.5·4-s + (−0.894 − 0.447i)5-s + (−0.696 + 0.119i)6-s + (0.0458 − 0.110i)7-s + 0.353·8-s + (0.942 − 0.333i)9-s + (−0.632 − 0.316i)10-s + (−1.54 + 0.639i)11-s + (−0.492 + 0.0845i)12-s + (−0.439 + 0.439i)13-s + (0.0324 − 0.0782i)14-s + (0.957 + 0.289i)15-s + 0.250·16-s + (−0.727 + 0.685i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 510 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0153i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 510 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0153i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(510\)    =    \(2 \cdot 3 \cdot 5 \cdot 17\)
Sign: $-0.999 - 0.0153i$
Analytic conductor: \(4.07237\)
Root analytic conductor: \(2.01801\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{510} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 510,\ (\ :1/2),\ -0.999 - 0.0153i)\)

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 \( 1 + (1.70 - 0.292i)T \)
5 \( 1 + (2 + i)T \)
17 \( 1 + (3 - 2.82i)T \)
good7 \( 1 + (-0.121 + 0.292i)T + (-4.94 - 4.94i)T^{2} \)
11 \( 1 + (5.12 - 2.12i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (1.58 - 1.58i)T - 13iT^{2} \)
19 \( 1 + (5.24 + 5.24i)T + 19iT^{2} \)
23 \( 1 + (2.70 + 1.12i)T + (16.2 + 16.2i)T^{2} \)
29 \( 1 + (4.12 + 1.70i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + (-0.707 + 1.70i)T + (-21.9 - 21.9i)T^{2} \)
37 \( 1 + (-1.29 - 3.12i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (3.12 + 7.53i)T + (-28.9 + 28.9i)T^{2} \)
43 \( 1 + 0.343iT - 43T^{2} \)
47 \( 1 + (0.171 - 0.171i)T - 47iT^{2} \)
53 \( 1 - 6.82iT - 53T^{2} \)
59 \( 1 + (3.58 + 3.58i)T + 59iT^{2} \)
61 \( 1 + (-10.9 + 4.53i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + (1.58 - 1.58i)T - 67iT^{2} \)
71 \( 1 + (-10.3 - 4.29i)T + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (-3.36 - 8.12i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (10.1 - 4.19i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + 0.343iT - 83T^{2} \)
89 \( 1 + 14.1iT - 89T^{2} \)
97 \( 1 + (8.53 - 3.53i)T + (68.5 - 68.5i)T^{2} \)
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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

−10.83916920615807432149305746024, −9.870562083984329573685736395240, −8.526047761184092982543663488265, −7.46766135492449091093923773863, −6.73285117076420804052892889610, −5.52328304204547191191630977329, −4.62736714202855921332598083703, −4.10503141518493789751057548896, −2.27395752973344080958453604511, 0, 2.39498216652732731623513901790, 3.74235296179905103760127929624, 4.85255032053737314025879887164, 5.66922392703823819566453844641, 6.64608398607402949666195965721, 7.59984017878792850319767191033, 8.266445061117708127835835429440, 10.08466558854534085552539742003, 10.74591360192952172032333264253

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