↑

Properties

Label 2-10-5.4-c25-0-5
Degree $2$
Conductor $10$
Sign $0.875 + 0.483i$
Analytic cond. $39.5996$
Root an. cond. $6.29282$
Motivic weight $25$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4.09e3i·2-s − 1.01e6i·3-s − 1.67e7·4-s + (−2.64e8 + 4.77e8i)5-s + 4.16e9·6-s − 1.16e10i·7-s − 6.87e10i·8-s − 1.86e11·9-s + (−1.95e12 − 1.08e12i)10-s − 5.94e12·11-s + 1.70e13i·12-s + 8.19e13i·13-s + 4.76e13·14-s + (4.85e14 + 2.68e14i)15-s + 2.81e14·16-s + 1.37e15i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 1.10i·3-s − 0.5·4-s + (−0.483 + 0.875i)5-s + 0.781·6-s − 0.317i·7-s − 0.353i·8-s − 0.220·9-s + (−0.618 − 0.341i)10-s − 0.571·11-s + 0.552i·12-s + 0.975i·13-s + 0.224·14-s + (0.966 + 0.534i)15-s + 0.250·16-s + 0.573i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.875 + 0.483i)\, \overline{\Lambda}(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & (0.875 + 0.483i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10\)    =    \(2 \cdot 5\)
Sign: $0.875 + 0.483i$
Analytic conductor: \(39.5996\)
Root analytic conductor: \(6.29282\)
Motivic weight: \(25\)
Rational: no
Arithmetic: yes
Character: $\chi_{10} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 10,\ (\ :25/2),\ 0.875 + 0.483i)\)

Particular Values

\(L(13)\) \(\approx\) \(1.420936347\)
\(L(\frac12)\) \(\approx\) \(1.420936347\)
\(L(\frac{27}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 4.09e3iT \)
5 \( 1 + (2.64e8 - 4.77e8i)T \)
good3 \( 1 + 1.01e6iT - 8.47e11T^{2} \)
7 \( 1 + 1.16e10iT - 1.34e21T^{2} \)
11 \( 1 + 5.94e12T + 1.08e26T^{2} \)
13 \( 1 - 8.19e13iT - 7.05e27T^{2} \)
17 \( 1 - 1.37e15iT - 5.77e30T^{2} \)
19 \( 1 + 4.66e15T + 9.30e31T^{2} \)
23 \( 1 + 9.70e16iT - 1.10e34T^{2} \)
29 \( 1 - 1.92e18T + 3.63e36T^{2} \)
31 \( 1 - 5.14e18T + 1.92e37T^{2} \)
37 \( 1 + 4.77e19iT - 1.60e39T^{2} \)
41 \( 1 - 1.13e20T + 2.08e40T^{2} \)
43 \( 1 + 9.41e18iT - 6.86e40T^{2} \)
47 \( 1 + 1.09e21iT - 6.34e41T^{2} \)
53 \( 1 + 6.11e20iT - 1.27e43T^{2} \)
59 \( 1 - 1.74e22T + 1.86e44T^{2} \)
61 \( 1 - 2.94e21T + 4.29e44T^{2} \)
67 \( 1 + 5.85e22iT - 4.48e45T^{2} \)
71 \( 1 + 2.27e23T + 1.91e46T^{2} \)
73 \( 1 - 2.69e23iT - 3.82e46T^{2} \)
79 \( 1 - 9.35e23T + 2.75e47T^{2} \)
83 \( 1 + 1.71e24iT - 9.48e47T^{2} \)
89 \( 1 - 2.89e24T + 5.42e48T^{2} \)
97 \( 1 - 1.13e24iT - 4.66e49T^{2} \)
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

−14.62278363480230416529980222755, −13.49696116452182446868366066979, −12.13260127987574583706123283590, −10.43717566877528569964570571321, −8.301116722459369901708817894112, −7.15929695459820799084733309581, −6.34551946189240074092743367917, −4.20018470983299807519438315883, −2.30257994124197262788793532345, −0.54616320793538293497226133873, 0.926754111223494282149123381427, 2.93161057185239182824843084224, 4.32530253280500704028038155227, 5.29521838830630186431335440827, 8.101994810451973977270613653402, 9.413290575838990661920539614793, 10.55129936806529882032030297340, 11.98907499084034898484831378348, 13.26709289038046915587445773562, 15.23883267053778326418099410524

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