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Properties

Label 2-26-13.6-c2-0-0
Degree $2$
Conductor $26$
Sign $0.428 - 0.903i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.366 + 1.36i)2-s + (−1.52 + 2.63i)3-s + (−1.73 + i)4-s + (4.79 − 4.79i)5-s + (−4.15 − 1.11i)6-s + (1.13 − 4.25i)7-s + (−2 − 1.99i)8-s + (−0.132 − 0.228i)9-s + (8.29 + 4.79i)10-s + (−13.8 + 3.71i)11-s − 6.08i·12-s + (1.84 − 12.8i)13-s + 6.22·14-s + (5.33 + 19.9i)15-s + (1.99 − 3.46i)16-s + (−20.9 + 12.1i)17-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (−0.507 + 0.878i)3-s + (−0.433 + 0.250i)4-s + (0.958 − 0.958i)5-s + (−0.692 − 0.185i)6-s + (0.162 − 0.607i)7-s + (−0.250 − 0.249i)8-s + (−0.0146 − 0.0254i)9-s + (0.829 + 0.479i)10-s + (−1.26 + 0.337i)11-s − 0.507i·12-s + (0.142 − 0.989i)13-s + 0.444·14-s + (0.355 + 1.32i)15-s + (0.124 − 0.216i)16-s + (−1.23 + 0.713i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.428 - 0.903i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.428 - 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(26\)    =    \(2 \cdot 13\)
Sign: $0.428 - 0.903i$
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: \((2,\ 26,\ (\ :1),\ 0.428 - 0.903i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.797163 + 0.504088i\)
\(L(\frac12)\) \(\approx\) \(0.797163 + 0.504088i\)
\(L(2)\) 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 \( 1 + (-1.84 + 12.8i)T \)
good3 \( 1 + (1.52 - 2.63i)T + (-4.5 - 7.79i)T^{2} \)
5 \( 1 + (-4.79 + 4.79i)T - 25iT^{2} \)
7 \( 1 + (-1.13 + 4.25i)T + (-42.4 - 24.5i)T^{2} \)
11 \( 1 + (13.8 - 3.71i)T + (104. - 60.5i)T^{2} \)
17 \( 1 + (20.9 - 12.1i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-25.4 - 6.82i)T + (312. + 180.5i)T^{2} \)
23 \( 1 + (5.44 + 3.14i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-11.1 + 19.2i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (8.59 - 8.59i)T - 961iT^{2} \)
37 \( 1 + (6.13 - 1.64i)T + (1.18e3 - 684.5i)T^{2} \)
41 \( 1 + (-18.8 - 70.4i)T + (-1.45e3 + 840.5i)T^{2} \)
43 \( 1 + (-26.9 + 15.5i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-7.65 - 7.65i)T + 2.20e3iT^{2} \)
53 \( 1 + 33.7T + 2.80e3T^{2} \)
59 \( 1 + (-9.77 + 36.4i)T + (-3.01e3 - 1.74e3i)T^{2} \)
61 \( 1 + (-11.5 - 19.9i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (27.8 + 103. i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + (-2.20 - 0.591i)T + (4.36e3 + 2.52e3i)T^{2} \)
73 \( 1 + (-38.1 - 38.1i)T + 5.32e3iT^{2} \)
79 \( 1 + 19.1T + 6.24e3T^{2} \)
83 \( 1 + (-34.7 + 34.7i)T - 6.88e3iT^{2} \)
89 \( 1 + (-3.47 + 0.930i)T + (6.85e3 - 3.96e3i)T^{2} \)
97 \( 1 + (24.3 + 6.51i)T + (8.14e3 + 4.70e3i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.27905456444789530368112102224, −16.21283475162627782321747990260, −15.42908404413254002415913698509, −13.65601544971397662628606055556, −12.84736499239125833691645533044, −10.67389132930586184320321905965, −9.649245623079041604553375536497, −7.938031072020424465250746126613, −5.68351322626094566961703263013, −4.67875794261568990535766948499, 2.39308500621309395742667048041, 5.59585237339799816418015853747, 7.00799684015701738839241920865, 9.296813006306228678138216179947, 10.81599139316149899503033178225, 11.87325995391701706659798274239, 13.27179097485458759740203114433, 14.06424256204560219922896248060, 15.74021765406459828305497481496, 17.83395785318610574556746966477

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