↑

Properties

Label 2-26-13.6-c2-0-1
Degree $2$
Conductor $26$
Sign $0.982 - 0.186i$
Analytic cond. $0.708448$
Root an. cond. $0.841693$
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  + (0.366 + 1.36i)2-s + (2.38 − 4.13i)3-s + (−1.73 + i)4-s + (−5.88 + 5.88i)5-s + (6.52 + 1.74i)6-s + (0.0922 − 0.344i)7-s + (−2 − 1.99i)8-s + (−6.90 − 11.9i)9-s + (−10.2 − 5.88i)10-s + (0.715 − 0.191i)11-s + 9.55i·12-s + (11.4 − 6.09i)13-s + 0.503·14-s + (10.2 + 38.4i)15-s + (1.99 − 3.46i)16-s + (−2.49 + 1.44i)17-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (0.795 − 1.37i)3-s + (−0.433 + 0.250i)4-s + (−1.17 + 1.17i)5-s + (1.08 + 0.291i)6-s + (0.0131 − 0.0491i)7-s + (−0.250 − 0.249i)8-s + (−0.767 − 1.32i)9-s + (−1.02 − 0.588i)10-s + (0.0650 − 0.0174i)11-s + 0.795i·12-s + (0.883 − 0.468i)13-s + 0.0359·14-s + (0.686 + 2.56i)15-s + (0.124 − 0.216i)16-s + (−0.146 + 0.0847i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.186i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.982 - 0.186i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(26\)    =    \(2 \cdot 13\)
Sign: $0.982 - 0.186i$
Analytic conductor: \(0.708448\)
Root analytic conductor: \(0.841693\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{26} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 26,\ (\ :1),\ 0.982 - 0.186i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.05012 + 0.0988281i\)
\(L(\frac12)\) \(\approx\) \(1.05012 + 0.0988281i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.366 - 1.36i)T \)
13 \( 1 + (-11.4 + 6.09i)T \)
good3 \( 1 + (-2.38 + 4.13i)T + (-4.5 - 7.79i)T^{2} \)
5 \( 1 + (5.88 - 5.88i)T - 25iT^{2} \)
7 \( 1 + (-0.0922 + 0.344i)T + (-42.4 - 24.5i)T^{2} \)
11 \( 1 + (-0.715 + 0.191i)T + (104. - 60.5i)T^{2} \)
17 \( 1 + (2.49 - 1.44i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (10.4 + 2.81i)T + (312. + 180.5i)T^{2} \)
23 \( 1 + (-19.8 - 11.4i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (15.2 - 26.3i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (-14.8 + 14.8i)T - 961iT^{2} \)
37 \( 1 + (40.2 - 10.7i)T + (1.18e3 - 684.5i)T^{2} \)
41 \( 1 + (6.66 + 24.8i)T + (-1.45e3 + 840.5i)T^{2} \)
43 \( 1 + (25.3 - 14.6i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-21.2 - 21.2i)T + 2.20e3iT^{2} \)
53 \( 1 + 85.8T + 2.80e3T^{2} \)
59 \( 1 + (-25.9 + 96.6i)T + (-3.01e3 - 1.74e3i)T^{2} \)
61 \( 1 + (7.73 + 13.3i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-18.0 - 67.2i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + (94.2 + 25.2i)T + (4.36e3 + 2.52e3i)T^{2} \)
73 \( 1 + (-50.9 - 50.9i)T + 5.32e3iT^{2} \)
79 \( 1 - 105.T + 6.24e3T^{2} \)
83 \( 1 + (27.2 - 27.2i)T - 6.88e3iT^{2} \)
89 \( 1 + (-82.1 + 22.0i)T + (6.85e3 - 3.96e3i)T^{2} \)
97 \( 1 + (19.6 + 5.26i)T + (8.14e3 + 4.70e3i)T^{2} \)
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

−17.57166920046896602316333551187, −15.65715941524906995168417983142, −14.78261821461939677309734361033, −13.73220501119800466669721514939, −12.56528193512319802957508645833, −11.08614423172769438702993265461, −8.534744674429230726170414161493, −7.51327801071358733594405478733, −6.59310787136014597901912063305, −3.34406627693961089623562420582, 3.72501728027450928871648736105, 4.73732082562542502302931571853, 8.408054940427337131476877541663, 9.141395244326484734021659091027, 10.71465053237868777596294007384, 11.99276097786504424211369109216, 13.45261149887121742294130488931, 14.96022603135642063969695271485, 15.82901656754505417239748933696, 16.80412866540898613233025778992

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