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Properties

Label 2-90e2-5.4-c1-0-29
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  + 0.680i·7-s + 1.68·11-s − 5.14i·13-s + 1.31i·17-s + 0.324·19-s + 3.78i·23-s − 8.64·29-s − 4.14·31-s + 1.35i·37-s + 7.15·41-s + 7.29i·43-s + 12.9i·47-s + 6.53·49-s − 8.83i·53-s + 8.81·59-s + ⋯
L(s)  = 1  + 0.257i·7-s + 0.506·11-s − 1.42i·13-s + 0.320i·17-s + 0.0744·19-s + 0.790i·23-s − 1.60·29-s − 0.744·31-s + 0.222i·37-s + 1.11·41-s + 1.11i·43-s + 1.89i·47-s + 0.933·49-s − 1.21i·53-s + 1.14·59-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

\(L(1)\) \(\approx\) \(1.881056914\)
\(L(\frac12)\) \(\approx\) \(1.881056914\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.680iT - 7T^{2} \)
11 \( 1 - 1.68T + 11T^{2} \)
13 \( 1 + 5.14iT - 13T^{2} \)
17 \( 1 - 1.31iT - 17T^{2} \)
19 \( 1 - 0.324T + 19T^{2} \)
23 \( 1 - 3.78iT - 23T^{2} \)
29 \( 1 + 8.64T + 29T^{2} \)
31 \( 1 + 4.14T + 31T^{2} \)
37 \( 1 - 1.35iT - 37T^{2} \)
41 \( 1 - 7.15T + 41T^{2} \)
43 \( 1 - 7.29iT - 43T^{2} \)
47 \( 1 - 12.9iT - 47T^{2} \)
53 \( 1 + 8.83iT - 53T^{2} \)
59 \( 1 - 8.81T + 59T^{2} \)
61 \( 1 - 9.97T + 61T^{2} \)
67 \( 1 + 4.17iT - 67T^{2} \)
71 \( 1 - 0.891T + 71T^{2} \)
73 \( 1 - 7.82iT - 73T^{2} \)
79 \( 1 + 9.65T + 79T^{2} \)
83 \( 1 - 4.85iT - 83T^{2} \)
89 \( 1 - 17.4T + 89T^{2} \)
97 \( 1 - 8.93iT - 97T^{2} \)
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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

−7.81190875720357166534233450971, −7.33829988013266934457120933986, −6.42967874003284346709151634624, −5.64885009549024536844659485277, −5.34862978835356816789005921479, −4.22496440812776641129967480611, −3.56459106546416513355164013992, −2.79681877245345020237916669954, −1.81539634196151852577741640086, −0.795574260795425459213994676177, 0.58316328410777859041765040154, 1.80064281706593016417293334848, 2.42449537443971183909600019890, 3.78659514640551352885272521061, 3.98429846998355552135113920589, 4.97108615540383654568896951220, 5.71008447027835181394078299180, 6.48053977186040622324789227023, 7.17765181354087052203458313104, 7.50656044098009958382272795397

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