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Properties

Label 2-90e2-5.4-c1-0-27
Degree $2$
Conductor $8100$
Sign $-0.447 - 0.894i$
Analytic cond. $64.6788$
Root an. cond. $8.04231$
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  + 2.73i·7-s + 1.73·11-s + 5.46i·13-s + 4.73i·17-s + 4.46·19-s + 3.46i·23-s + 7.73·29-s + 5.92·31-s + 6.19i·37-s − 11.1·41-s + 3.26i·43-s + 1.26i·47-s − 0.464·49-s − 7.26i·53-s + 7.73·59-s + ⋯
L(s)  = 1  + 1.03i·7-s + 0.522·11-s + 1.51i·13-s + 1.14i·17-s + 1.02·19-s + 0.722i·23-s + 1.43·29-s + 1.06·31-s + 1.01i·37-s − 1.74·41-s + 0.498i·43-s + 0.184i·47-s − 0.0663·49-s − 0.998i·53-s + 1.00·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8100\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{2}\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(64.6788\)
Root analytic conductor: \(8.04231\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{8100} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 8100,\ (\ :1/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.080741594\)
\(L(\frac12)\) \(\approx\) \(2.080741594\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 - 2.73iT - 7T^{2} \)
11 \( 1 - 1.73T + 11T^{2} \)
13 \( 1 - 5.46iT - 13T^{2} \)
17 \( 1 - 4.73iT - 17T^{2} \)
19 \( 1 - 4.46T + 19T^{2} \)
23 \( 1 - 3.46iT - 23T^{2} \)
29 \( 1 - 7.73T + 29T^{2} \)
31 \( 1 - 5.92T + 31T^{2} \)
37 \( 1 - 6.19iT - 37T^{2} \)
41 \( 1 + 11.1T + 41T^{2} \)
43 \( 1 - 3.26iT - 43T^{2} \)
47 \( 1 - 1.26iT - 47T^{2} \)
53 \( 1 + 7.26iT - 53T^{2} \)
59 \( 1 - 7.73T + 59T^{2} \)
61 \( 1 + 4T + 61T^{2} \)
67 \( 1 + 6.39iT - 67T^{2} \)
71 \( 1 + 11.1T + 71T^{2} \)
73 \( 1 + 0.196iT - 73T^{2} \)
79 \( 1 - 14.3T + 79T^{2} \)
83 \( 1 + 15.1iT - 83T^{2} \)
89 \( 1 + 5.19T + 89T^{2} \)
97 \( 1 + 0.732iT - 97T^{2} \)
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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

−8.256519959179155696621198121407, −7.29834426769673725009459997296, −6.41685055121913014898748685960, −6.25476909054397872385685394344, −5.17553230044184579934163596062, −4.62954806244140513454880081954, −3.71411330143282053361577324111, −2.97206610722112583193702064000, −1.96559437160426288413765850720, −1.28545367012722313446893825016, 0.56443098735172459396247610140, 1.11737561732758244511718546820, 2.59190134347928429341286424265, 3.19946138318874930605478935298, 4.02307549549852819407144278773, 4.83428259123963801455432860241, 5.40528842672002556285390491265, 6.31804795685642687403488504562, 7.03677004925769630920027695132, 7.49109308774663033486318363518

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