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Properties

Label 2-3100-31.30-c2-0-25
Degree $2$
Conductor $3100$
Sign $0.220 - 0.975i$
Analytic cond. $84.4688$
Root an. cond. $9.19069$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.99i·3-s − 11.3·7-s − 15.9·9-s − 13.5i·11-s − 0.886i·13-s − 12.3i·17-s − 37.2·19-s − 56.7i·21-s − 24.9i·23-s − 34.5i·27-s + 24.1i·29-s + (−6.84 + 30.2i)31-s + 67.4·33-s − 50.7i·37-s + 4.42·39-s + ⋯
L(s)  = 1  + 1.66i·3-s − 1.62·7-s − 1.76·9-s − 1.22i·11-s − 0.0681i·13-s − 0.724i·17-s − 1.96·19-s − 2.70i·21-s − 1.08i·23-s − 1.27i·27-s + 0.832i·29-s + (−0.220 + 0.975i)31-s + 2.04·33-s − 1.37i·37-s + 0.113·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.220 - 0.975i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.220 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3100\)    =    \(2^{2} \cdot 5^{2} \cdot 31\)
Sign: $0.220 - 0.975i$
Analytic conductor: \(84.4688\)
Root analytic conductor: \(9.19069\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{3100} (1301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3100,\ (\ :1),\ 0.220 - 0.975i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.8035348848\)
\(L(\frac12)\) \(\approx\) \(0.8035348848\)
\(L(2)\) 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 \( 1 \)
31 \( 1 + (6.84 - 30.2i)T \)
good3 \( 1 - 4.99iT - 9T^{2} \)
7 \( 1 + 11.3T + 49T^{2} \)
11 \( 1 + 13.5iT - 121T^{2} \)
13 \( 1 + 0.886iT - 169T^{2} \)
17 \( 1 + 12.3iT - 289T^{2} \)
19 \( 1 + 37.2T + 361T^{2} \)
23 \( 1 + 24.9iT - 529T^{2} \)
29 \( 1 - 24.1iT - 841T^{2} \)
37 \( 1 + 50.7iT - 1.36e3T^{2} \)
41 \( 1 + 32.4T + 1.68e3T^{2} \)
43 \( 1 - 56.8iT - 1.84e3T^{2} \)
47 \( 1 - 40.4T + 2.20e3T^{2} \)
53 \( 1 - 18.1iT - 2.80e3T^{2} \)
59 \( 1 - 15.8T + 3.48e3T^{2} \)
61 \( 1 - 61.7iT - 3.72e3T^{2} \)
67 \( 1 - 38.6T + 4.48e3T^{2} \)
71 \( 1 + 98.4T + 5.04e3T^{2} \)
73 \( 1 + 63.3iT - 5.32e3T^{2} \)
79 \( 1 + 73.1iT - 6.24e3T^{2} \)
83 \( 1 - 95.8iT - 6.88e3T^{2} \)
89 \( 1 - 162. iT - 7.92e3T^{2} \)
97 \( 1 - 177.T + 9.40e3T^{2} \)
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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

−8.981895574955191908769040900323, −8.342217881746901657533110170496, −6.96166986105980107617203598259, −6.25711772702179962770817499312, −5.63752773719930847729703953734, −4.67796085512161573511354182074, −3.91536742974852754238263804789, −3.24284372737862113664599706570, −2.60150891294061656190570453671, −0.43262869483434382382259770655, 0.39215686366877123751209081628, 1.80643057973286264190423323044, 2.32019978882781511291571567243, 3.43869264071391568908378060172, 4.36233787805007327325513808351, 5.75904201242282176120295025882, 6.30121855216089902342998459977, 6.85525944945359037773944879194, 7.39875401743385979100324015416, 8.246429830547981390074091031384

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