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Properties

Label 2-2900-145.12-c1-0-30
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

\(L(1)\) \(\approx\) \(3.821700787\)
\(L(\frac12)\) \(\approx\) \(3.821700787\)
\(L(\frac{3}{2})\) 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

−8.832108090120655528067996696484, −8.141635750837282047256467906763, −7.54142888378582857965088368656, −6.74537245880160912790002252802, −5.90376959094209741932740468820, −4.53266856081058321422160217968, −3.97479501581250687665889891626, −3.18958004639994488183275886274, −2.16403486061382678326940435482, −1.47297713313040359440255281849, 1.07591965830360021909877852512, 2.20264610824037035704218842963, 3.00486701034638394198440042694, 3.92559003431725013662724150319, 4.29298510667502899823626733539, 5.90164053841896186441881926857, 6.36289614563444726031162493844, 7.59113994150492451332549220475, 8.220680652504881876638470835496, 8.460472921610819603971839198511

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