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Properties

Label 2-2900-145.12-c1-0-11
Degree $2$
Conductor $2900$
Sign $0.883 - 0.467i$
Analytic cond. $23.1566$
Root an. cond. $4.81213$
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  − 2.96·3-s + (−0.336 − 0.336i)7-s + 5.81·9-s + (1.26 + 1.26i)11-s + (−4.51 − 4.51i)13-s + 6.03i·17-s + (−2.26 + 2.26i)19-s + (1 + i)21-s + (6.37 − 6.37i)23-s − 8.36·27-s + (−4.04 − 3.55i)29-s + (6.33 + 6.33i)31-s + (−3.74 − 3.74i)33-s − 8.26·37-s + (13.4 + 13.4i)39-s + ⋯
L(s)  = 1  − 1.71·3-s + (−0.127 − 0.127i)7-s + 1.93·9-s + (0.380 + 0.380i)11-s + (−1.25 − 1.25i)13-s + 1.46i·17-s + (−0.518 + 0.518i)19-s + (0.218 + 0.218i)21-s + (1.33 − 1.33i)23-s − 1.60·27-s + (−0.751 − 0.660i)29-s + (1.13 + 1.13i)31-s + (−0.652 − 0.652i)33-s − 1.35·37-s + (2.14 + 2.14i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.883 - 0.467i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.883 - 0.467i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2900\)    =    \(2^{2} \cdot 5^{2} \cdot 29\)
Sign: $0.883 - 0.467i$
Analytic conductor: \(23.1566\)
Root analytic conductor: \(4.81213\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2900} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2900,\ (\ :1/2),\ 0.883 - 0.467i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6667537115\)
\(L(\frac12)\) \(\approx\) \(0.6667537115\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
29 \( 1 + (4.04 + 3.55i)T \)
good3 \( 1 + 2.96T + 3T^{2} \)
7 \( 1 + (0.336 + 0.336i)T + 7iT^{2} \)
11 \( 1 + (-1.26 - 1.26i)T + 11iT^{2} \)
13 \( 1 + (4.51 + 4.51i)T + 13iT^{2} \)
17 \( 1 - 6.03iT - 17T^{2} \)
19 \( 1 + (2.26 - 2.26i)T - 19iT^{2} \)
23 \( 1 + (-6.37 + 6.37i)T - 23iT^{2} \)
31 \( 1 + (-6.33 - 6.33i)T + 31iT^{2} \)
37 \( 1 + 8.26T + 37T^{2} \)
41 \( 1 + (0.216 - 0.216i)T - 41iT^{2} \)
43 \( 1 + 3.84T + 43T^{2} \)
47 \( 1 + 1.42T + 47T^{2} \)
53 \( 1 + (-1.45 + 1.45i)T - 53iT^{2} \)
59 \( 1 - 8.38iT - 59T^{2} \)
61 \( 1 + (9.11 + 9.11i)T + 61iT^{2} \)
67 \( 1 + (3.31 - 3.31i)T - 67iT^{2} \)
71 \( 1 + 2.81iT - 71T^{2} \)
73 \( 1 - 9.10iT - 73T^{2} \)
79 \( 1 + (-5.84 + 5.84i)T - 79iT^{2} \)
83 \( 1 + (-8.79 + 8.79i)T - 83iT^{2} \)
89 \( 1 + (5.30 - 5.30i)T - 89iT^{2} \)
97 \( 1 - 0.750T + 97T^{2} \)
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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

−8.796827272430778210295770517601, −7.953455375967136514757883294472, −7.01211860996163841535012254756, −6.51649083867857375956455559285, −5.76141237551070470419519401919, −4.99644531925774751152407562881, −4.45898128094621489660772103301, −3.30450079341590002406031560338, −1.88967045192332575580742944777, −0.65225618879834045069438482255, 0.45921265906117607496877577872, 1.72072097018483113492010666339, 3.04145012680180652221687951199, 4.36675918440186927008109376500, 4.94037039345464239777209192811, 5.52875636776754094090030022637, 6.51685299579207903961158012522, 6.97846237364409440275445076809, 7.56203769609638574445467383099, 9.087840992151067466421568492522

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