Properties

Label 2-2900-5.4-c1-0-11
Degree $2$
Conductor $2900$
Sign $0.447 - 0.894i$
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  + 3·9-s − 2·11-s + 2i·13-s + 2·19-s + 8i·23-s − 29-s + 2·31-s − 4i·37-s − 10·41-s − 4i·43-s + 12i·47-s + 7·49-s + 6i·53-s + 12·59-s − 10·61-s + ⋯
L(s)  = 1  + 9-s − 0.603·11-s + 0.554i·13-s + 0.458·19-s + 1.66i·23-s − 0.185·29-s + 0.359·31-s − 0.657i·37-s − 1.56·41-s − 0.609i·43-s + 1.75i·47-s + 49-s + 0.824i·53-s + 1.56·59-s − 1.28·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2900\)    =    \(2^{2} \cdot 5^{2} \cdot 29\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(23.1566\)
Root analytic conductor: \(4.81213\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2900} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2900,\ (\ :1/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.704403235\)
\(L(\frac12)\) \(\approx\) \(1.704403235\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
good3 \( 1 - 3T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 - 8iT - 23T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 + 4iT - 37T^{2} \)
41 \( 1 + 10T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 - 12iT - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 - 12T + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 - 12iT - 67T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + 12iT - 73T^{2} \)
79 \( 1 + 2T + 79T^{2} \)
83 \( 1 - 4iT - 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 - 8iT - 97T^{2} \)
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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

−9.052978364136340422138182232645, −7.981080650961969339112582302446, −7.40905058891638774434677971101, −6.77931181155221894837203349012, −5.76455817118344703199740287192, −5.04591397138533902146460590538, −4.15731252643609686278314349118, −3.36132623144193292438649194868, −2.18101251717164426773249743024, −1.18601639715493187244629271629, 0.59435553238787811240763273639, 1.90406125190320875997852166940, 2.92133659661270515027024579327, 3.88751859682147661393167948392, 4.80493460468917129426563953448, 5.41279940944279165696937535638, 6.54379005075471290013237555818, 7.02680600150139887846831327106, 8.000137497253049560981281616718, 8.453442722646973244918741045268

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