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Properties

Label 2-2900-5.4-c1-0-7
Degree $2$
Conductor $2900$
Sign $-0.894 + 0.447i$
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  + 2.40i·3-s + 4.71i·7-s − 2.79·9-s + 4.05·11-s − 4.25i·13-s + 5.20i·17-s − 5.68·19-s − 11.3·21-s − 4.74i·23-s + 0.488i·27-s − 29-s − 10.0·31-s + 9.75i·33-s + 7.01i·37-s + 10.2·39-s + ⋯
L(s)  = 1  + 1.39i·3-s + 1.78i·7-s − 0.932·9-s + 1.22·11-s − 1.17i·13-s + 1.26i·17-s − 1.30·19-s − 2.47·21-s − 0.988i·23-s + 0.0940i·27-s − 0.185·29-s − 1.79·31-s + 1.69i·33-s + 1.15i·37-s + 1.64·39-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2900\)    =    \(2^{2} \cdot 5^{2} \cdot 29\)
Sign: $-0.894 + 0.447i$
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: \((2,\ 2900,\ (\ :1/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.249178101\)
\(L(\frac12)\) \(\approx\) \(1.249178101\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.40iT - 3T^{2} \)
7 \( 1 - 4.71iT - 7T^{2} \)
11 \( 1 - 4.05T + 11T^{2} \)
13 \( 1 + 4.25iT - 13T^{2} \)
17 \( 1 - 5.20iT - 17T^{2} \)
19 \( 1 + 5.68T + 19T^{2} \)
23 \( 1 + 4.74iT - 23T^{2} \)
31 \( 1 + 10.0T + 31T^{2} \)
37 \( 1 - 7.01iT - 37T^{2} \)
41 \( 1 + 2.17T + 41T^{2} \)
43 \( 1 + 1.68iT - 43T^{2} \)
47 \( 1 - 6.89iT - 47T^{2} \)
53 \( 1 - 12.6iT - 53T^{2} \)
59 \( 1 - 12.8T + 59T^{2} \)
61 \( 1 + 0.542T + 61T^{2} \)
67 \( 1 + 13.2iT - 67T^{2} \)
71 \( 1 - 2.84T + 71T^{2} \)
73 \( 1 - 4.66iT - 73T^{2} \)
79 \( 1 - 6.55T + 79T^{2} \)
83 \( 1 + 7.14iT - 83T^{2} \)
89 \( 1 + 4.04T + 89T^{2} \)
97 \( 1 - 9.36iT - 97T^{2} \)
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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

−9.166274826246530667915745575436, −8.637846302703049882209257164437, −8.126031468049188426451241994319, −6.64421109530771610188451640190, −5.93860304127892731807178435409, −5.38536212240183920354818370255, −4.45713324194958399064689830219, −3.73655635249111459118000754049, −2.86490589065628799656806703466, −1.79514694153556935368593493404, 0.38996824749756194098393151625, 1.42358781796656939372007091289, 2.12119938568194358194011006924, 3.75406982640020695192626210243, 4.09401205542830403165032798160, 5.32884527639053432238740753755, 6.53736680404637082772351897464, 6.98253060110691030447942247149, 7.21067885196292193941799270693, 8.122080725171013442369583778330

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