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Properties

Label 2-510-15.2-c1-0-31
Degree $2$
Conductor $510$
Sign $-0.678 - 0.734i$
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  + (0.707 − 0.707i)2-s + (−1.33 − 1.10i)3-s − 1.00i·4-s + (−1.73 − 1.41i)5-s + (−1.72 + 0.158i)6-s + (−1 − i)7-s + (−0.707 − 0.707i)8-s + (0.548 + 2.94i)9-s + (−2.22 + 0.224i)10-s − 1.41i·11-s + (−1.10 + 1.33i)12-s + (−1.77 + 1.77i)13-s − 1.41·14-s + (0.741 + 3.80i)15-s − 1.00·16-s + (−0.707 + 0.707i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.769 − 0.639i)3-s − 0.500i·4-s + (−0.774 − 0.632i)5-s + (−0.704 + 0.0648i)6-s + (−0.377 − 0.377i)7-s + (−0.250 − 0.250i)8-s + (0.182 + 0.983i)9-s + (−0.703 + 0.0710i)10-s − 0.426i·11-s + (−0.319 + 0.384i)12-s + (−0.492 + 0.492i)13-s − 0.377·14-s + (0.191 + 0.981i)15-s − 0.250·16-s + (−0.171 + 0.171i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 510 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.678 - 0.734i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 510 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.678 - 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(510\)    =    \(2 \cdot 3 \cdot 5 \cdot 17\)
Sign: $-0.678 - 0.734i$
Analytic conductor: \(4.07237\)
Root analytic conductor: \(2.01801\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{510} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 510,\ (\ :1/2),\ -0.678 - 0.734i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.184209 + 0.421141i\)
\(L(\frac12)\) \(\approx\) \(0.184209 + 0.421141i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 + (1.33 + 1.10i)T \)
5 \( 1 + (1.73 + 1.41i)T \)
17 \( 1 + (0.707 - 0.707i)T \)
good7 \( 1 + (1 + i)T + 7iT^{2} \)
11 \( 1 + 1.41iT - 11T^{2} \)
13 \( 1 + (1.77 - 1.77i)T - 13iT^{2} \)
19 \( 1 - 2.55iT - 19T^{2} \)
23 \( 1 + (1.73 + 1.73i)T + 23iT^{2} \)
29 \( 1 + 6.61T + 29T^{2} \)
31 \( 1 - 3T + 31T^{2} \)
37 \( 1 + (6.89 + 6.89i)T + 37iT^{2} \)
41 \( 1 - 5.51iT - 41T^{2} \)
43 \( 1 + (2.55 - 2.55i)T - 43iT^{2} \)
47 \( 1 + (-3.53 + 3.53i)T - 47iT^{2} \)
53 \( 1 + (7.95 + 7.95i)T + 53iT^{2} \)
59 \( 1 - 0.317T + 59T^{2} \)
61 \( 1 - 8.34T + 61T^{2} \)
67 \( 1 + (10.4 + 10.4i)T + 67iT^{2} \)
71 \( 1 + 4.56iT - 71T^{2} \)
73 \( 1 + (-0.325 + 0.325i)T - 73iT^{2} \)
79 \( 1 + 6.89iT - 79T^{2} \)
83 \( 1 + (-5.65 - 5.65i)T + 83iT^{2} \)
89 \( 1 + 15.8T + 89T^{2} \)
97 \( 1 + (-4.12 - 4.12i)T + 97iT^{2} \)
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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

−10.66466656804334778285944341054, −9.661038014040644132053794175908, −8.458173987688287729862471638224, −7.48705196030855485589496057813, −6.56671444921690141175562560302, −5.52552874018617899963061333229, −4.57504293098799731818556029166, −3.56266465591680741722574874914, −1.81431465159565246598310684671, −0.24737273176838044711413021771, 2.88322885306328497154168500534, 3.93187731070568069855682431074, 4.89760928859676721822706434408, 5.85894137597247479706344884104, 6.82820859940183318233057475090, 7.53823747548402388610719697606, 8.797034237751693503664899277688, 9.815101399243288764326182258930, 10.67264570330562903361943778031, 11.58444857349305494513223098206

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