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Properties

Label 2-3100-31.30-c2-0-14
Degree $2$
Conductor $3100$
Sign $0.603 + 0.797i$
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  + 5.52i·3-s − 8.23·7-s − 21.5·9-s + 13.8i·11-s + 4.20i·13-s + 14.5i·17-s − 9.90·19-s − 45.4i·21-s + 42.3i·23-s − 69.1i·27-s + 28.8i·29-s + (−18.7 − 24.7i)31-s − 76.6·33-s + 7.59i·37-s − 23.2·39-s + ⋯
L(s)  = 1  + 1.84i·3-s − 1.17·7-s − 2.39·9-s + 1.26i·11-s + 0.323i·13-s + 0.856i·17-s − 0.521·19-s − 2.16i·21-s + 1.84i·23-s − 2.56i·27-s + 0.994i·29-s + (−0.603 − 0.797i)31-s − 2.32·33-s + 0.205i·37-s − 0.595·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.603 + 0.797i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.603 + 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3100\)    =    \(2^{2} \cdot 5^{2} \cdot 31\)
Sign: $0.603 + 0.797i$
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: \((2,\ 3100,\ (\ :1),\ 0.603 + 0.797i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.7305314946\)
\(L(\frac12)\) \(\approx\) \(0.7305314946\)
\(L(2)\) 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 \( 1 + (18.7 + 24.7i)T \)
good3 \( 1 - 5.52iT - 9T^{2} \)
7 \( 1 + 8.23T + 49T^{2} \)
11 \( 1 - 13.8iT - 121T^{2} \)
13 \( 1 - 4.20iT - 169T^{2} \)
17 \( 1 - 14.5iT - 289T^{2} \)
19 \( 1 + 9.90T + 361T^{2} \)
23 \( 1 - 42.3iT - 529T^{2} \)
29 \( 1 - 28.8iT - 841T^{2} \)
37 \( 1 - 7.59iT - 1.36e3T^{2} \)
41 \( 1 + 21.0T + 1.68e3T^{2} \)
43 \( 1 - 78.7iT - 1.84e3T^{2} \)
47 \( 1 + 76.1T + 2.20e3T^{2} \)
53 \( 1 + 62.4iT - 2.80e3T^{2} \)
59 \( 1 + 32.7T + 3.48e3T^{2} \)
61 \( 1 - 60.6iT - 3.72e3T^{2} \)
67 \( 1 - 43.3T + 4.48e3T^{2} \)
71 \( 1 - 67.0T + 5.04e3T^{2} \)
73 \( 1 - 29.7iT - 5.32e3T^{2} \)
79 \( 1 + 15.9iT - 6.24e3T^{2} \)
83 \( 1 - 81.7iT - 6.88e3T^{2} \)
89 \( 1 + 79.5iT - 7.92e3T^{2} \)
97 \( 1 - 151.T + 9.40e3T^{2} \)
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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

−9.466825904671555387809285555316, −8.726797921081573840504534872408, −7.78983564022545517278636080496, −6.74884555572638393769704434059, −6.01095456112980933881938051691, −5.15761646534820072852139108356, −4.46579430110990273670375804240, −3.65530144079436756225383998881, −3.18480227789441104176520678594, −1.88417226148532750022435666682, 0.25048878153027642019039372536, 0.62938598139372318723257227304, 2.04158609780891475655150583594, 2.85195787097122071427981998029, 3.51029305093725424112828553518, 5.04787410608981611215523701299, 6.04771166390033480530380601587, 6.39867771041487780493438618244, 7.00447354115098318826951435029, 7.82418243808594139890836184880

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