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Properties

Label 2-3100-31.30-c2-0-29
Degree $2$
Conductor $3100$
Sign $-0.981 + 0.193i$
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  + 4.48i·3-s + 3.21·7-s − 11.0·9-s + 18.8i·11-s − 13.8i·13-s + 13.8i·17-s + 8.23·19-s + 14.4i·21-s + 21.0i·23-s − 9.41i·27-s + 22.1i·29-s + (30.4 − 5.99i)31-s − 84.4·33-s + 6.01i·37-s + 62.2·39-s + ⋯
L(s)  = 1  + 1.49i·3-s + 0.459·7-s − 1.23·9-s + 1.71i·11-s − 1.06i·13-s + 0.813i·17-s + 0.433·19-s + 0.687i·21-s + 0.916i·23-s − 0.348i·27-s + 0.762i·29-s + (0.981 − 0.193i)31-s − 2.55·33-s + 0.162i·37-s + 1.59·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 + 0.193i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.981 + 0.193i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3100\)    =    \(2^{2} \cdot 5^{2} \cdot 31\)
Sign: $-0.981 + 0.193i$
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: \((2,\ 3100,\ (\ :1),\ -0.981 + 0.193i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.829624967\)
\(L(\frac12)\) \(\approx\) \(1.829624967\)
\(L(2)\) 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 \( 1 + (-30.4 + 5.99i)T \)
good3 \( 1 - 4.48iT - 9T^{2} \)
7 \( 1 - 3.21T + 49T^{2} \)
11 \( 1 - 18.8iT - 121T^{2} \)
13 \( 1 + 13.8iT - 169T^{2} \)
17 \( 1 - 13.8iT - 289T^{2} \)
19 \( 1 - 8.23T + 361T^{2} \)
23 \( 1 - 21.0iT - 529T^{2} \)
29 \( 1 - 22.1iT - 841T^{2} \)
37 \( 1 - 6.01iT - 1.36e3T^{2} \)
41 \( 1 - 22.3T + 1.68e3T^{2} \)
43 \( 1 + 5.93iT - 1.84e3T^{2} \)
47 \( 1 - 48.9T + 2.20e3T^{2} \)
53 \( 1 - 52.0iT - 2.80e3T^{2} \)
59 \( 1 + 63.3T + 3.48e3T^{2} \)
61 \( 1 + 85.7iT - 3.72e3T^{2} \)
67 \( 1 - 62.1T + 4.48e3T^{2} \)
71 \( 1 - 52.1T + 5.04e3T^{2} \)
73 \( 1 - 62.7iT - 5.32e3T^{2} \)
79 \( 1 - 6.84iT - 6.24e3T^{2} \)
83 \( 1 - 109. iT - 6.88e3T^{2} \)
89 \( 1 - 29.8iT - 7.92e3T^{2} \)
97 \( 1 - 139.T + 9.40e3T^{2} \)
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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

−9.235322101398386048401981107077, −8.146222933229868452512176203454, −7.64375826562479509193627858364, −6.63416596383588125834227871021, −5.50819341794353534314774738637, −5.02334875976012600749916848857, −4.29415032676404529309581321419, −3.59459459050336568911864578564, −2.59501912106330245752346078218, −1.37255978860930283599374224034, 0.44211629961582924811705983696, 1.17289872732463271680262950081, 2.25344689618912756139680872095, 3.00927024474832995981147870456, 4.20549960466875783699717581879, 5.21296677194237583786680167490, 6.17675906073707176107263232052, 6.53162572160991527773972747250, 7.43513510342595822569539753030, 8.033060290481230409638150915787

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