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Properties

Label 2-810-135.122-c1-0-11
Degree $2$
Conductor $810$
Sign $0.407 + 0.913i$
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  + (−0.819 − 0.573i)2-s + (0.342 + 0.939i)4-s + (2.23 + 0.138i)5-s + (0.660 + 1.41i)7-s + (0.258 − 0.965i)8-s + (−1.74 − 1.39i)10-s + (−2.49 − 2.97i)11-s + (−2.53 − 3.62i)13-s + (0.271 − 1.53i)14-s + (−0.766 + 0.642i)16-s + (−1.52 − 5.69i)17-s + (5.43 − 3.13i)19-s + (0.633 + 2.14i)20-s + (0.338 + 3.87i)22-s + (1.84 + 0.862i)23-s + ⋯
L(s)  = 1  + (−0.579 − 0.405i)2-s + (0.171 + 0.469i)4-s + (0.998 + 0.0618i)5-s + (0.249 + 0.535i)7-s + (0.0915 − 0.341i)8-s + (−0.553 − 0.440i)10-s + (−0.753 − 0.897i)11-s + (−0.703 − 1.00i)13-s + (0.0725 − 0.411i)14-s + (−0.191 + 0.160i)16-s + (−0.370 − 1.38i)17-s + (1.24 − 0.719i)19-s + (0.141 + 0.479i)20-s + (0.0722 + 0.825i)22-s + (0.385 + 0.179i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.407 + 0.913i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.407 + 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $0.407 + 0.913i$
Analytic conductor: \(6.46788\)
Root analytic conductor: \(2.54320\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{810} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 810,\ (\ :1/2),\ 0.407 + 0.913i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09672 - 0.711671i\)
\(L(\frac12)\) \(\approx\) \(1.09672 - 0.711671i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.819 + 0.573i)T \)
3 \( 1 \)
5 \( 1 + (-2.23 - 0.138i)T \)
good7 \( 1 + (-0.660 - 1.41i)T + (-4.49 + 5.36i)T^{2} \)
11 \( 1 + (2.49 + 2.97i)T + (-1.91 + 10.8i)T^{2} \)
13 \( 1 + (2.53 + 3.62i)T + (-4.44 + 12.2i)T^{2} \)
17 \( 1 + (1.52 + 5.69i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-5.43 + 3.13i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.84 - 0.862i)T + (14.7 + 17.6i)T^{2} \)
29 \( 1 + (-0.651 - 3.69i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (-4.48 + 1.63i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (-5.27 + 1.41i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (1.36 + 0.240i)T + (38.5 + 14.0i)T^{2} \)
43 \( 1 + (0.741 - 8.47i)T + (-42.3 - 7.46i)T^{2} \)
47 \( 1 + (-2.58 + 1.20i)T + (30.2 - 36.0i)T^{2} \)
53 \( 1 + (4.58 + 4.58i)T + 53iT^{2} \)
59 \( 1 + (0.286 + 0.240i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-3.34 - 1.21i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-9.14 + 6.40i)T + (22.9 - 62.9i)T^{2} \)
71 \( 1 + (-13.8 - 7.99i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (15.8 + 4.24i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-0.515 + 0.0908i)T + (74.2 - 27.0i)T^{2} \)
83 \( 1 + (6.06 - 8.66i)T + (-28.3 - 77.9i)T^{2} \)
89 \( 1 + (-2.59 - 4.49i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (11.4 + 1.00i)T + (95.5 + 16.8i)T^{2} \)
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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

−9.888729887151830341785304304345, −9.462474505552592981263005617706, −8.538361197422116628247429097642, −7.68086919438309272265377456376, −6.73718548784206088579254588621, −5.47525407539395104074006943091, −5.02443112586853991753584432027, −2.96416996385704323975687292578, −2.58165002108678989647358273318, −0.844635566270336958957269934764, 1.45403985829212994019511097804, 2.46981137634636580355701547190, 4.27819829446151300066200739437, 5.18753886297057327330452421998, 6.16418504607219292152342361024, 7.03708424254281772781850052955, 7.78713747940432436473061291325, 8.749139035930516951172628074083, 9.744711756093697865991404071261, 10.09041301125649370642896120730

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