↑

Properties

Label 2-3100-31.30-c2-0-13
Degree $2$
Conductor $3100$
Sign $-0.986 + 0.164i$
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  + 3.71i·3-s + 1.89·7-s − 4.78·9-s − 3.40i·11-s + 11.5i·13-s − 23.9i·17-s + 5.61·19-s + 7.03i·21-s + 5.50i·23-s + 15.6i·27-s + 50.5i·29-s + (−30.5 + 5.11i)31-s + 12.6·33-s − 58.2i·37-s − 42.9·39-s + ⋯
L(s)  = 1  + 1.23i·3-s + 0.270·7-s − 0.531·9-s − 0.309i·11-s + 0.890i·13-s − 1.40i·17-s + 0.295·19-s + 0.335i·21-s + 0.239i·23-s + 0.579i·27-s + 1.74i·29-s + (−0.986 + 0.164i)31-s + 0.383·33-s − 1.57i·37-s − 1.10·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 + 0.164i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.986 + 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3100\)    =    \(2^{2} \cdot 5^{2} \cdot 31\)
Sign: $-0.986 + 0.164i$
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: \((2,\ 3100,\ (\ :1),\ -0.986 + 0.164i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.121489000\)
\(L(\frac12)\) \(\approx\) \(1.121489000\)
\(L(2)\) 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 \( 1 + (30.5 - 5.11i)T \)
good3 \( 1 - 3.71iT - 9T^{2} \)
7 \( 1 - 1.89T + 49T^{2} \)
11 \( 1 + 3.40iT - 121T^{2} \)
13 \( 1 - 11.5iT - 169T^{2} \)
17 \( 1 + 23.9iT - 289T^{2} \)
19 \( 1 - 5.61T + 361T^{2} \)
23 \( 1 - 5.50iT - 529T^{2} \)
29 \( 1 - 50.5iT - 841T^{2} \)
37 \( 1 + 58.2iT - 1.36e3T^{2} \)
41 \( 1 - 35.1T + 1.68e3T^{2} \)
43 \( 1 + 17.2iT - 1.84e3T^{2} \)
47 \( 1 + 69.6T + 2.20e3T^{2} \)
53 \( 1 - 14.5iT - 2.80e3T^{2} \)
59 \( 1 - 45.0T + 3.48e3T^{2} \)
61 \( 1 - 89.1iT - 3.72e3T^{2} \)
67 \( 1 - 129.T + 4.48e3T^{2} \)
71 \( 1 + 115.T + 5.04e3T^{2} \)
73 \( 1 - 97.3iT - 5.32e3T^{2} \)
79 \( 1 - 50.8iT - 6.24e3T^{2} \)
83 \( 1 - 59.9iT - 6.88e3T^{2} \)
89 \( 1 - 137. iT - 7.92e3T^{2} \)
97 \( 1 + 139.T + 9.40e3T^{2} \)
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

−9.253194110206262879293963266100, −8.387173591198508829432896770038, −7.28795379112546289672690936666, −6.83154406005914347352997579971, −5.46180082585443916726692499412, −5.14836153414606522920709816139, −4.20406234486728903681179675021, −3.57399922379714382615773563196, −2.59730220648253339509879377434, −1.30945098196553910951896914855, 0.24975924882105701496195250713, 1.39327313797817064526969442730, 2.09075586694231429662058570016, 3.16083494076072377328037667149, 4.20036445268325733970541737074, 5.16150643124552691635239308906, 6.15894489999624060999762290462, 6.49949607269576261909474514299, 7.59886280646568307399028428128, 7.948390642514966183853951799770

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