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Properties

Label 2-810-45.38-c1-0-0
Degree $2$
Conductor $810$
Sign $-0.993 + 0.116i$
Analytic cond. $6.46788$
Root an. cond. $2.54320$
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  + (−0.965 + 0.258i)2-s + (0.866 − 0.499i)4-s + (−1.48 + 1.67i)5-s + (1.13 + 4.23i)7-s + (−0.707 + 0.707i)8-s + (1 − 2i)10-s + (−1.22 − 0.707i)11-s + (−1.5 + 5.59i)13-s + (−2.19 − 3.79i)14-s + (0.500 − 0.866i)16-s + (−4.38 − 4.38i)17-s − 3.19i·19-s + (−0.448 + 2.19i)20-s + (1.36 + 0.366i)22-s + (4.82 + 1.29i)23-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (0.433 − 0.249i)4-s + (−0.663 + 0.748i)5-s + (0.428 + 1.59i)7-s + (−0.249 + 0.249i)8-s + (0.316 − 0.632i)10-s + (−0.369 − 0.213i)11-s + (−0.416 + 1.55i)13-s + (−0.585 − 1.01i)14-s + (0.125 − 0.216i)16-s + (−1.06 − 1.06i)17-s − 0.733i·19-s + (−0.100 + 0.489i)20-s + (0.291 + 0.0780i)22-s + (1.00 + 0.269i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.116i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 + 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $-0.993 + 0.116i$
Analytic conductor: \(6.46788\)
Root analytic conductor: \(2.54320\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{810} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 810,\ (\ :1/2),\ -0.993 + 0.116i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0272428 - 0.464195i\)
\(L(\frac12)\) \(\approx\) \(0.0272428 - 0.464195i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 - 0.258i)T \)
3 \( 1 \)
5 \( 1 + (1.48 - 1.67i)T \)
good7 \( 1 + (-1.13 - 4.23i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (1.22 + 0.707i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.5 - 5.59i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (4.38 + 4.38i)T + 17iT^{2} \)
19 \( 1 + 3.19iT - 19T^{2} \)
23 \( 1 + (-4.82 - 1.29i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (2.82 - 4.89i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.09 + 5.36i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.19 + 5.19i)T - 37iT^{2} \)
41 \( 1 + (1.10 - 0.637i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (7.09 - 1.90i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (0.776 - 0.208i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (2.26 - 2.26i)T - 53iT^{2} \)
59 \( 1 + (1.48 + 2.56i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.90 - 3.29i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.73 + 1.53i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 10.1iT - 71T^{2} \)
73 \( 1 + (-4 - 4i)T + 73iT^{2} \)
79 \( 1 + (1.90 + 1.09i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.67 - 13.7i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 - 1.69T + 89T^{2} \)
97 \( 1 + (2.19 + 8.19i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.97198362913417040083742700426, −9.397323088529696794338795466139, −9.157168389030551423035436895009, −8.234459712831550346289606172661, −7.20171058746760317078866402843, −6.66859083760180134595092108880, −5.47757809144565341884020161723, −4.51225192286054917731186737211, −2.86950422060410658083066517169, −2.12404445878264181503493599296, 0.28606555979945308265088549071, 1.52218045641225269061131155342, 3.29300709713050881533559743038, 4.26842147955868680747878816627, 5.16853088284084811854809883495, 6.59831667116584366619464817823, 7.69079025315669649677353078641, 7.925267612946678806197207813742, 8.832364647478116309447489156623, 10.01212567928172834057001961219

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