↑

Properties

Label 2-510-15.2-c1-0-3
Degree $2$
Conductor $510$
Sign $0.526 - 0.850i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 + 0.707i)2-s + (−1.58 + 0.705i)3-s − 1.00i·4-s + (−2.23 + 0.0974i)5-s + (0.619 − 1.61i)6-s + (−3.25 − 3.25i)7-s + (0.707 + 0.707i)8-s + (2.00 − 2.23i)9-s + (1.51 − 1.64i)10-s + 3.90i·11-s + (0.705 + 1.58i)12-s + (2.89 − 2.89i)13-s + 4.60·14-s + (3.46 − 1.73i)15-s − 1.00·16-s + (−0.707 + 0.707i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−0.913 + 0.407i)3-s − 0.500i·4-s + (−0.999 + 0.0435i)5-s + (0.252 − 0.660i)6-s + (−1.23 − 1.23i)7-s + (0.250 + 0.250i)8-s + (0.668 − 0.744i)9-s + (0.477 − 0.521i)10-s + 1.17i·11-s + (0.203 + 0.456i)12-s + (0.802 − 0.802i)13-s + 1.23·14-s + (0.894 − 0.446i)15-s − 0.250·16-s + (−0.171 + 0.171i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 510 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.526 - 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 510 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.526 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(510\)    =    \(2 \cdot 3 \cdot 5 \cdot 17\)
Sign: $0.526 - 0.850i$
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: \((2,\ 510,\ (\ :1/2),\ 0.526 - 0.850i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.430624 + 0.239969i\)
\(L(\frac12)\) \(\approx\) \(0.430624 + 0.239969i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.58 - 0.705i)T \)
5 \( 1 + (2.23 - 0.0974i)T \)
17 \( 1 + (0.707 - 0.707i)T \)
good7 \( 1 + (3.25 + 3.25i)T + 7iT^{2} \)
11 \( 1 - 3.90iT - 11T^{2} \)
13 \( 1 + (-2.89 + 2.89i)T - 13iT^{2} \)
19 \( 1 - 5.28iT - 19T^{2} \)
23 \( 1 + (-1.08 - 1.08i)T + 23iT^{2} \)
29 \( 1 + 1.81T + 29T^{2} \)
31 \( 1 - 7.47T + 31T^{2} \)
37 \( 1 + (3.81 + 3.81i)T + 37iT^{2} \)
41 \( 1 + 3.03iT - 41T^{2} \)
43 \( 1 + (3.66 - 3.66i)T - 43iT^{2} \)
47 \( 1 + (-3.79 + 3.79i)T - 47iT^{2} \)
53 \( 1 + (-8.80 - 8.80i)T + 53iT^{2} \)
59 \( 1 - 8.47T + 59T^{2} \)
61 \( 1 - 10.1T + 61T^{2} \)
67 \( 1 + (-4.29 - 4.29i)T + 67iT^{2} \)
71 \( 1 - 3.19iT - 71T^{2} \)
73 \( 1 + (-7.14 + 7.14i)T - 73iT^{2} \)
79 \( 1 + 3.97iT - 79T^{2} \)
83 \( 1 + (-3.62 - 3.62i)T + 83iT^{2} \)
89 \( 1 + 7.98T + 89T^{2} \)
97 \( 1 + (-9.40 - 9.40i)T + 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.62063262399240875021292599038, −10.33211832188666828537263361722, −9.504983733754558738719573598690, −8.201751864414330495381204084879, −7.20308758341980175521611799749, −6.70540063319784072508430837271, −5.60524640352090101098994102279, −4.26627134089209604858274498242, −3.60277955450784797593376067841, −0.812951138505086513704700070289, 0.61528929760268119891162084348, 2.61871055434685051003336613880, 3.75391900985423541395985075046, 5.16361424678102267760192431016, 6.38765918920632002460551808969, 6.90129249124186279233130779770, 8.380445371880188960053866610264, 8.842365713011215166601565564343, 9.952467628634500801911774043503, 11.12384346288352627858323963081

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