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Properties

Label 2-810-45.34-c1-0-5
Degree $2$
Conductor $810$
Sign $0.993 + 0.114i$
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  + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (−2.23 + 0.133i)5-s + (−3.46 − 2i)7-s − 0.999i·8-s + (1.99 + i)10-s + (−2.5 + 4.33i)11-s + (2.59 − 1.5i)13-s + (1.99 + 3.46i)14-s + (−0.5 + 0.866i)16-s + i·17-s + 6·19-s + (−1.23 − 1.86i)20-s + (4.33 − 2.5i)22-s + (−0.866 + 0.5i)23-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.998 + 0.0599i)5-s + (−1.30 − 0.755i)7-s − 0.353i·8-s + (0.632 + 0.316i)10-s + (−0.753 + 1.30i)11-s + (0.720 − 0.416i)13-s + (0.534 + 0.925i)14-s + (−0.125 + 0.216i)16-s + 0.242i·17-s + 1.37·19-s + (−0.275 − 0.417i)20-s + (0.923 − 0.533i)22-s + (−0.180 + 0.104i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.114i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.114i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $0.993 + 0.114i$
Analytic conductor: \(6.46788\)
Root analytic conductor: \(2.54320\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{810} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 810,\ (\ :1/2),\ 0.993 + 0.114i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.701738 - 0.0402472i\)
\(L(\frac12)\) \(\approx\) \(0.701738 - 0.0402472i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 \)
5 \( 1 + (2.23 - 0.133i)T \)
good7 \( 1 + (3.46 + 2i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.5 - 4.33i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.59 + 1.5i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - iT - 17T^{2} \)
19 \( 1 - 6T + 19T^{2} \)
23 \( 1 + (0.866 - 0.5i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.5 + 7.79i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.5 - 4.33i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 + (1 + 1.73i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.866 - 0.5i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-11.2 - 6.5i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (-2 - 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4 - 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.46 - 2i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 - 2iT - 73T^{2} \)
79 \( 1 + (-4.5 + 7.79i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.46 - 2i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 14T + 89T^{2} \)
97 \( 1 + (-8.66 - 5i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.19555094637498824679483833363, −9.620337672341380705805123084841, −8.528391657347319940429168123871, −7.53246042984797751155590429272, −7.20411872753956148714103964363, −6.05398288366295894766321756352, −4.54301729585982935162908124043, −3.62128093636301350650063934230, −2.74156307016556199733257937344, −0.802072659257077854794451982425, 0.67387621334735811500964198602, 2.85713809642824637269763380975, 3.55459662778451630863306637633, 5.15245874821136837747524187016, 6.03979768306878775210534390471, 6.84652879546366066660465707507, 7.82835925124504963353508608802, 8.630747143905917356925403220439, 9.178784425528731829032300811439, 10.18326384083530758732037907837

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