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Properties

Label 2-810-135.122-c1-0-17
Degree $2$
Conductor $810$
Sign $-0.924 - 0.381i$
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 + (0.463 − 2.18i)5-s + (−1.78 − 3.83i)7-s + (0.258 − 0.965i)8-s + (−1.63 + 1.52i)10-s + (−3.34 − 3.98i)11-s + (−0.215 − 0.307i)13-s + (−0.734 + 4.16i)14-s + (−0.766 + 0.642i)16-s + (1.58 + 5.90i)17-s + (−2.93 + 1.69i)19-s + (2.21 − 0.313i)20-s + (0.453 + 5.18i)22-s + (1.01 + 0.472i)23-s + ⋯
L(s)  = 1  + (−0.579 − 0.405i)2-s + (0.171 + 0.469i)4-s + (0.207 − 0.978i)5-s + (−0.675 − 1.44i)7-s + (0.0915 − 0.341i)8-s + (−0.516 + 0.482i)10-s + (−1.00 − 1.20i)11-s + (−0.0597 − 0.0853i)13-s + (−0.196 + 1.11i)14-s + (−0.191 + 0.160i)16-s + (0.383 + 1.43i)17-s + (−0.672 + 0.388i)19-s + (0.495 − 0.0699i)20-s + (0.0966 + 1.10i)22-s + (0.211 + 0.0984i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.924 - 0.381i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.924 - 0.381i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $-0.924 - 0.381i$
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: \((2,\ 810,\ (\ :1/2),\ -0.924 - 0.381i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0991386 + 0.499609i\)
\(L(\frac12)\) \(\approx\) \(0.0991386 + 0.499609i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.463 + 2.18i)T \)
good7 \( 1 + (1.78 + 3.83i)T + (-4.49 + 5.36i)T^{2} \)
11 \( 1 + (3.34 + 3.98i)T + (-1.91 + 10.8i)T^{2} \)
13 \( 1 + (0.215 + 0.307i)T + (-4.44 + 12.2i)T^{2} \)
17 \( 1 + (-1.58 - 5.90i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (2.93 - 1.69i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.01 - 0.472i)T + (14.7 + 17.6i)T^{2} \)
29 \( 1 + (-1.75 - 9.97i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (3.35 - 1.22i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (-5.63 + 1.51i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (4.77 + 0.841i)T + (38.5 + 14.0i)T^{2} \)
43 \( 1 + (-0.570 + 6.52i)T + (-42.3 - 7.46i)T^{2} \)
47 \( 1 + (-6.65 + 3.10i)T + (30.2 - 36.0i)T^{2} \)
53 \( 1 + (6.39 + 6.39i)T + 53iT^{2} \)
59 \( 1 + (2.30 + 1.93i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (7.18 + 2.61i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-0.199 + 0.139i)T + (22.9 - 62.9i)T^{2} \)
71 \( 1 + (0.0530 + 0.0306i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-6.38 - 1.71i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (5.15 - 0.909i)T + (74.2 - 27.0i)T^{2} \)
83 \( 1 + (-2.73 + 3.91i)T + (-28.3 - 77.9i)T^{2} \)
89 \( 1 + (4.63 + 8.03i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (15.6 + 1.36i)T + (95.5 + 16.8i)T^{2} \)
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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

−9.961717381420606854400458952789, −8.841165264948051663925892203258, −8.268630408576225880771035712514, −7.42403631237463012606091293450, −6.32143976178263239151031708911, −5.31912736652242879842041585557, −4.02841969550506147613423260540, −3.25334745533510946046213905756, −1.53017101833610254742246938993, −0.29703709832105443623943547079, 2.36433220036074567903724355579, 2.76700180384748269940010781461, 4.67517565683422895296795479704, 5.72280997548444441609298070375, 6.43098271836976724208407696201, 7.33656739420198685327395682696, 8.038683850167669878460039027352, 9.399148203610060075264515271513, 9.569056100548080844202579974806, 10.50062091444022883625226633288

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