Properties

Label 2-90e2-5.4-c1-0-8
Degree $2$
Conductor $8100$
Sign $-0.894 - 0.447i$
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  + 2.85i·7-s + 1.85·11-s + 0.854i·13-s − 1.14i·17-s − 2·19-s − 4.85i·23-s − 3.70·29-s + 2.70·31-s + 5.85i·37-s − 11.5·41-s + 0.854i·43-s + 6.70i·47-s − 1.14·49-s + 4.85i·53-s − 1.14·59-s + ⋯
L(s)  = 1  + 1.07i·7-s + 0.559·11-s + 0.236i·13-s − 0.277i·17-s − 0.458·19-s − 1.01i·23-s − 0.688·29-s + 0.486·31-s + 0.962i·37-s − 1.80·41-s + 0.130i·43-s + 0.978i·47-s − 0.163·49-s + 0.666i·53-s − 0.149·59-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(8100\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{2}\)
Sign: $-0.894 - 0.447i$
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: \((2,\ 8100,\ (\ :1/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9003343668\)
\(L(\frac12)\) \(\approx\) \(0.9003343668\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.85iT - 7T^{2} \)
11 \( 1 - 1.85T + 11T^{2} \)
13 \( 1 - 0.854iT - 13T^{2} \)
17 \( 1 + 1.14iT - 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 + 4.85iT - 23T^{2} \)
29 \( 1 + 3.70T + 29T^{2} \)
31 \( 1 - 2.70T + 31T^{2} \)
37 \( 1 - 5.85iT - 37T^{2} \)
41 \( 1 + 11.5T + 41T^{2} \)
43 \( 1 - 0.854iT - 43T^{2} \)
47 \( 1 - 6.70iT - 47T^{2} \)
53 \( 1 - 4.85iT - 53T^{2} \)
59 \( 1 + 1.14T + 59T^{2} \)
61 \( 1 - 0.854T + 61T^{2} \)
67 \( 1 - 7iT - 67T^{2} \)
71 \( 1 - 9T + 71T^{2} \)
73 \( 1 - 2.70iT - 73T^{2} \)
79 \( 1 + 11.7T + 79T^{2} \)
83 \( 1 - 6.70iT - 83T^{2} \)
89 \( 1 - 12T + 89T^{2} \)
97 \( 1 - 10iT - 97T^{2} \)
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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

−8.349913735327346582958026400749, −7.39341629588402712157193459307, −6.58171354082493854533482156141, −6.17673729161197891453750300823, −5.31491082978398197103130200962, −4.66939831252681329352168391057, −3.86437782856170392515247448501, −2.91083784088174534269873659026, −2.24077725466411603929953345379, −1.26392896710841313195533349435, 0.21202165684284673663922051451, 1.32576602300885447917966658340, 2.16157800486398701998417842853, 3.50263357319905574674097347266, 3.75230822145319863106847380486, 4.68377260700630546865821227095, 5.41387041403962100526961760154, 6.25379993171981873418718185233, 6.90472079087382840377418103470, 7.47229270850675541311716452211

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