Properties

Label 2-90e2-5.4-c1-0-2
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  + 3.85i·7-s + 4.85·11-s + 5.85i·13-s − 7.85i·17-s − 2·19-s + 1.85i·23-s − 9.70·29-s − 10.7·31-s + 0.854i·37-s − 8.56·41-s + 5.85i·43-s − 6.70i·47-s − 7.85·49-s − 1.85i·53-s + 7.85·59-s + ⋯
L(s)  = 1  + 1.45i·7-s + 1.46·11-s + 1.62i·13-s − 1.90i·17-s − 0.458·19-s + 0.386i·23-s − 1.80·29-s − 1.92·31-s + 0.140i·37-s − 1.33·41-s + 0.892i·43-s − 0.978i·47-s − 1.12·49-s − 0.254i·53-s + 1.02·59-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

\(L(1)\) \(\approx\) \(0.3107400376\)
\(L(\frac12)\) \(\approx\) \(0.3107400376\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.85iT - 7T^{2} \)
11 \( 1 - 4.85T + 11T^{2} \)
13 \( 1 - 5.85iT - 13T^{2} \)
17 \( 1 + 7.85iT - 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 - 1.85iT - 23T^{2} \)
29 \( 1 + 9.70T + 29T^{2} \)
31 \( 1 + 10.7T + 31T^{2} \)
37 \( 1 - 0.854iT - 37T^{2} \)
41 \( 1 + 8.56T + 41T^{2} \)
43 \( 1 - 5.85iT - 43T^{2} \)
47 \( 1 + 6.70iT - 47T^{2} \)
53 \( 1 + 1.85iT - 53T^{2} \)
59 \( 1 - 7.85T + 59T^{2} \)
61 \( 1 + 5.85T + 61T^{2} \)
67 \( 1 + 7iT - 67T^{2} \)
71 \( 1 + 9T + 71T^{2} \)
73 \( 1 - 10.7iT - 73T^{2} \)
79 \( 1 - 1.70T + 79T^{2} \)
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

−8.487372730219815324383021977344, −7.12795612254734157630958758630, −7.07872415719619938744988618280, −6.10507684723132614489041191300, −5.49508888204037693567148359799, −4.76326206378914541540946852650, −3.93504148877096053167845362831, −3.18707393718463949473412585535, −2.07648768522129582501150223552, −1.65984269310057518664778583726, 0.06971180995740946256409079588, 1.21561918558594517682565781011, 1.91068786781128630019585703597, 3.45183191474502671831870754192, 3.74478911123238913731921522301, 4.34882145851907382488452969453, 5.49569798373039337955769926734, 6.02268752831127715358587620906, 6.84537107549693777926594658619, 7.38840383305697968489477486405

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