Properties

Label 2-3100-31.30-c2-0-16
Degree $2$
Conductor $3100$
Sign $-0.994 - 0.107i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 0.880i·3-s − 4.35·7-s + 8.22·9-s + 18.9i·11-s + 3.17i·13-s + 14.3i·17-s − 14.4·19-s − 3.83i·21-s + 10.0i·23-s + 15.1i·27-s − 9.65i·29-s + (30.8 + 3.33i)31-s − 16.6·33-s + 31.1i·37-s − 2.79·39-s + ⋯
L(s)  = 1  + 0.293i·3-s − 0.622·7-s + 0.913·9-s + 1.71i·11-s + 0.244i·13-s + 0.845i·17-s − 0.758·19-s − 0.182i·21-s + 0.435i·23-s + 0.561i·27-s − 0.332i·29-s + (0.994 + 0.107i)31-s − 0.504·33-s + 0.842i·37-s − 0.0716·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.107i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.994 - 0.107i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3100\)    =    \(2^{2} \cdot 5^{2} \cdot 31\)
Sign: $-0.994 - 0.107i$
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: \((2,\ 3100,\ (\ :1),\ -0.994 - 0.107i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.097096512\)
\(L(\frac12)\) \(\approx\) \(1.097096512\)
\(L(2)\) 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 \( 1 + (-30.8 - 3.33i)T \)
good3 \( 1 - 0.880iT - 9T^{2} \)
7 \( 1 + 4.35T + 49T^{2} \)
11 \( 1 - 18.9iT - 121T^{2} \)
13 \( 1 - 3.17iT - 169T^{2} \)
17 \( 1 - 14.3iT - 289T^{2} \)
19 \( 1 + 14.4T + 361T^{2} \)
23 \( 1 - 10.0iT - 529T^{2} \)
29 \( 1 + 9.65iT - 841T^{2} \)
37 \( 1 - 31.1iT - 1.36e3T^{2} \)
41 \( 1 - 28.9T + 1.68e3T^{2} \)
43 \( 1 + 58.0iT - 1.84e3T^{2} \)
47 \( 1 + 67.3T + 2.20e3T^{2} \)
53 \( 1 - 89.1iT - 2.80e3T^{2} \)
59 \( 1 - 103.T + 3.48e3T^{2} \)
61 \( 1 + 32.7iT - 3.72e3T^{2} \)
67 \( 1 + 44.0T + 4.48e3T^{2} \)
71 \( 1 + 121.T + 5.04e3T^{2} \)
73 \( 1 + 49.1iT - 5.32e3T^{2} \)
79 \( 1 + 62.3iT - 6.24e3T^{2} \)
83 \( 1 + 124. iT - 6.88e3T^{2} \)
89 \( 1 - 156. iT - 7.92e3T^{2} \)
97 \( 1 - 48.5T + 9.40e3T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.034775208610552410394691316844, −8.092239827095521557526909695006, −7.27834497246404676022855055494, −6.71087408509334403264860575604, −5.97796422473362524389288908393, −4.73620563884637903072550609642, −4.36022096253796576482563169791, −3.46654598239429575034913195087, −2.25056012720589380119568906560, −1.43250990496267385959365337755, 0.25932168475968051068088200204, 1.14539610044726998747900593972, 2.51002648558415690292483056093, 3.28669294357498208271309247473, 4.17543042133197885419006835393, 5.11423892114505643582130534334, 6.10265835641233371744204425094, 6.55390150684745100998906842605, 7.37181581033465229509483753887, 8.247784520454396183692280588769

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