Properties

Label 2-30e2-5.4-c1-0-0
Degree $2$
Conductor $900$
Sign $-0.894 - 0.447i$
Analytic cond. $7.18653$
Root an. cond. $2.68077$
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  + i·7-s − 6·11-s + 5i·13-s − 6i·17-s − 5·19-s + 6i·23-s − 6·29-s − 31-s − 2i·37-s i·43-s + 6i·47-s + 6·49-s + 12i·53-s − 6·59-s − 13·61-s + ⋯
L(s)  = 1  + 0.377i·7-s − 1.80·11-s + 1.38i·13-s − 1.45i·17-s − 1.14·19-s + 1.25i·23-s − 1.11·29-s − 0.179·31-s − 0.328i·37-s − 0.152i·43-s + 0.875i·47-s + 0.857·49-s + 1.64i·53-s − 0.781·59-s − 1.66·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.894 - 0.447i$
Analytic conductor: \(7.18653\)
Root analytic conductor: \(2.68077\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.101794 + 0.431210i\)
\(L(\frac12)\) \(\approx\) \(0.101794 + 0.431210i\)
\(L(\frac{3}{2})\) 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 \( 1 - iT - 7T^{2} \)
11 \( 1 + 6T + 11T^{2} \)
13 \( 1 - 5iT - 13T^{2} \)
17 \( 1 + 6iT - 17T^{2} \)
19 \( 1 + 5T + 19T^{2} \)
23 \( 1 - 6iT - 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + T + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + iT - 43T^{2} \)
47 \( 1 - 6iT - 47T^{2} \)
53 \( 1 - 12iT - 53T^{2} \)
59 \( 1 + 6T + 59T^{2} \)
61 \( 1 + 13T + 61T^{2} \)
67 \( 1 + 11iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 2iT - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 - 6iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 7iT - 97T^{2} \)
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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

−10.57035585260762501504220656045, −9.435590974838287656801479702750, −9.009248093768973114603293531539, −7.77257173406062130709802700467, −7.26442280926462827992290063716, −6.06321439432679930640564944032, −5.20963005992701011660948774043, −4.33422707656086795869755599092, −2.94284828645428034475902093026, −1.98586538432933420897346775636, 0.19230769664162608861973081968, 2.12064936219419524615740731141, 3.22525608976952906489068533611, 4.38823556100248134338029319870, 5.41916014624861283158612246916, 6.16490166397990327785233876558, 7.36767190819819267159735340736, 8.127989041971554567784853107520, 8.643413251964296609821545157696, 10.20148949022245829906742717178

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