↑

Properties

Label 2-10-5.4-c25-0-6
Degree $2$
Conductor $10$
Sign $0.385 - 0.922i$
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 + 1.16e5i·3-s − 1.67e7·4-s + (5.03e8 + 2.10e8i)5-s − 4.77e8·6-s + 1.34e10i·7-s − 6.87e10i·8-s + 8.33e11·9-s + (−8.61e11 + 2.06e12i)10-s + 4.67e11·11-s − 1.95e12i·12-s − 9.38e13i·13-s − 5.50e13·14-s + (−2.45e13 + 5.87e13i)15-s + 2.81e14·16-s − 6.14e14i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.126i·3-s − 0.5·4-s + (0.922 + 0.385i)5-s − 0.0896·6-s + 0.366i·7-s − 0.353i·8-s + 0.983·9-s + (−0.272 + 0.652i)10-s + 0.0449·11-s − 0.0633i·12-s − 1.11i·13-s − 0.259·14-s + (−0.0488 + 0.116i)15-s + 0.250·16-s − 0.255i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.385 - 0.922i)\, \overline{\Lambda}(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & (0.385 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10\)    =    \(2 \cdot 5\)
Sign: $0.385 - 0.922i$
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: \((2,\ 10,\ (\ :25/2),\ 0.385 - 0.922i)\)

Particular Values

\(L(13)\) \(\approx\) \(2.657630395\)
\(L(\frac12)\) \(\approx\) \(2.657630395\)
\(L(\frac{27}{2})\) 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 \( 1 + (-5.03e8 - 2.10e8i)T \)
good3 \( 1 - 1.16e5iT - 8.47e11T^{2} \)
7 \( 1 - 1.34e10iT - 1.34e21T^{2} \)
11 \( 1 - 4.67e11T + 1.08e26T^{2} \)
13 \( 1 + 9.38e13iT - 7.05e27T^{2} \)
17 \( 1 + 6.14e14iT - 5.77e30T^{2} \)
19 \( 1 - 1.66e16T + 9.30e31T^{2} \)
23 \( 1 + 1.60e17iT - 1.10e34T^{2} \)
29 \( 1 - 1.05e18T + 3.63e36T^{2} \)
31 \( 1 + 3.00e18T + 1.92e37T^{2} \)
37 \( 1 - 3.36e19iT - 1.60e39T^{2} \)
41 \( 1 - 5.09e19T + 2.08e40T^{2} \)
43 \( 1 - 2.92e20iT - 6.86e40T^{2} \)
47 \( 1 - 9.77e20iT - 6.34e41T^{2} \)
53 \( 1 - 5.26e21iT - 1.27e43T^{2} \)
59 \( 1 + 4.10e20T + 1.86e44T^{2} \)
61 \( 1 + 5.44e21T + 4.29e44T^{2} \)
67 \( 1 + 6.32e22iT - 4.48e45T^{2} \)
71 \( 1 + 1.81e22T + 1.91e46T^{2} \)
73 \( 1 + 3.48e23iT - 3.82e46T^{2} \)
79 \( 1 + 3.43e23T + 2.75e47T^{2} \)
83 \( 1 - 6.85e23iT - 9.48e47T^{2} \)
89 \( 1 - 4.29e24T + 5.42e48T^{2} \)
97 \( 1 - 2.90e24iT - 4.66e49T^{2} \)
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

−15.15036448090625916485312890993, −13.86806918253161700377461439850, −12.60664193059192387359028949656, −10.42955008492104969447372260371, −9.326254895854953247997609874271, −7.56404070140333593713023968730, −6.17075791885175860540054389483, −4.90114900828829700861381718655, −2.92260221209362246998612110831, −1.07636374296660905240897561343, 1.03565808735497307518916924106, 1.94639112265335441476403229797, 3.83396290039262547692095450107, 5.33108950814270971923282967586, 7.17294077156464376694582134752, 9.191304857496838673278713235886, 10.12172037427502374463888288771, 11.78065917612870711449837815893, 13.18175539727991984880952253315, 14.05759997960596910013538895354

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