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Properties

Label 2-3100-31.30-c2-0-49
Degree $2$
Conductor $3100$
Sign $0.911 - 0.410i$
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  + 2.53i·3-s − 7.38·7-s + 2.55·9-s + 14.4i·11-s − 7.77i·13-s − 18.0i·17-s − 22.9·19-s − 18.7i·21-s − 21.4i·23-s + 29.3i·27-s + 18.8i·29-s + (28.2 − 12.7i)31-s − 36.6·33-s − 17.5i·37-s + 19.7·39-s + ⋯
L(s)  = 1  + 0.846i·3-s − 1.05·7-s + 0.283·9-s + 1.31i·11-s − 0.597i·13-s − 1.06i·17-s − 1.20·19-s − 0.893i·21-s − 0.931i·23-s + 1.08i·27-s + 0.649i·29-s + (0.911 − 0.410i)31-s − 1.10·33-s − 0.473i·37-s + 0.506·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 - 0.410i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.911 - 0.410i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3100\)    =    \(2^{2} \cdot 5^{2} \cdot 31\)
Sign: $0.911 - 0.410i$
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: \((2,\ 3100,\ (\ :1),\ 0.911 - 0.410i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.563334455\)
\(L(\frac12)\) \(\approx\) \(1.563334455\)
\(L(2)\) 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 \( 1 + (-28.2 + 12.7i)T \)
good3 \( 1 - 2.53iT - 9T^{2} \)
7 \( 1 + 7.38T + 49T^{2} \)
11 \( 1 - 14.4iT - 121T^{2} \)
13 \( 1 + 7.77iT - 169T^{2} \)
17 \( 1 + 18.0iT - 289T^{2} \)
19 \( 1 + 22.9T + 361T^{2} \)
23 \( 1 + 21.4iT - 529T^{2} \)
29 \( 1 - 18.8iT - 841T^{2} \)
37 \( 1 + 17.5iT - 1.36e3T^{2} \)
41 \( 1 - 53.5T + 1.68e3T^{2} \)
43 \( 1 + 16.6iT - 1.84e3T^{2} \)
47 \( 1 + 30.2T + 2.20e3T^{2} \)
53 \( 1 + 41.6iT - 2.80e3T^{2} \)
59 \( 1 - 22.7T + 3.48e3T^{2} \)
61 \( 1 + 34.3iT - 3.72e3T^{2} \)
67 \( 1 - 0.949T + 4.48e3T^{2} \)
71 \( 1 - 88.5T + 5.04e3T^{2} \)
73 \( 1 - 8.94iT - 5.32e3T^{2} \)
79 \( 1 - 19.0iT - 6.24e3T^{2} \)
83 \( 1 - 108. iT - 6.88e3T^{2} \)
89 \( 1 + 79.2iT - 7.92e3T^{2} \)
97 \( 1 - 57.8T + 9.40e3T^{2} \)
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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

−8.767145859489779878056720254538, −7.80587695489627023245750825463, −6.92150805251185192255094391507, −6.48215406912867788310156639084, −5.33283107747284537720533558331, −4.59824003256627106041991677254, −3.99309213195531623789386336838, −3.00485582477419390334555300405, −2.13574123944862340519465040558, −0.53027444255198908666929954545, 0.68143306764291626184175548594, 1.70887402064417851544047506082, 2.75203694909647650890306117570, 3.68204628630345631678645757074, 4.43406822338128240810847462928, 5.82504593901430538079643074206, 6.31297613997686141611584049457, 6.76259251938366515408387233314, 7.78651832538210535464753052171, 8.348051635927535930024352150544

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