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Properties

Label 2-26-13.7-c2-0-2
Degree $2$
Conductor $26$
Sign $0.999 + 0.0193i$
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  + (1.36 − 0.366i)2-s + (−0.866 + 1.5i)3-s + (1.73 − i)4-s + (−1.73 − 1.73i)5-s + (−0.633 + 2.36i)6-s + (−8.96 − 2.40i)7-s + (1.99 − 2i)8-s + (3 + 5.19i)9-s + (−2.99 − 1.73i)10-s + (1.96 + 7.33i)11-s + 3.46i·12-s + (3.92 − 12.3i)13-s − 13.1·14-s + (4.09 − 1.09i)15-s + (1.99 − 3.46i)16-s + (14.3 − 8.25i)17-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (−0.288 + 0.5i)3-s + (0.433 − 0.250i)4-s + (−0.346 − 0.346i)5-s + (−0.105 + 0.394i)6-s + (−1.28 − 0.343i)7-s + (0.249 − 0.250i)8-s + (0.333 + 0.577i)9-s + (−0.299 − 0.173i)10-s + (0.178 + 0.666i)11-s + 0.288i·12-s + (0.302 − 0.953i)13-s − 0.937·14-s + (0.273 − 0.0732i)15-s + (0.124 − 0.216i)16-s + (0.841 − 0.485i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0193i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0193i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(26\)    =    \(2 \cdot 13\)
Sign: $0.999 + 0.0193i$
Analytic conductor: \(0.708448\)
Root analytic conductor: \(0.841693\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{26} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 26,\ (\ :1),\ 0.999 + 0.0193i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.12035 - 0.0108555i\)
\(L(\frac12)\) \(\approx\) \(1.12035 - 0.0108555i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36 + 0.366i)T \)
13 \( 1 + (-3.92 + 12.3i)T \)
good3 \( 1 + (0.866 - 1.5i)T + (-4.5 - 7.79i)T^{2} \)
5 \( 1 + (1.73 + 1.73i)T + 25iT^{2} \)
7 \( 1 + (8.96 + 2.40i)T + (42.4 + 24.5i)T^{2} \)
11 \( 1 + (-1.96 - 7.33i)T + (-104. + 60.5i)T^{2} \)
17 \( 1 + (-14.3 + 8.25i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (4.10 - 15.3i)T + (-312. - 180.5i)T^{2} \)
23 \( 1 + (-21.1 - 12.2i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (27.3 - 47.3i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (40.6 + 40.6i)T + 961iT^{2} \)
37 \( 1 + (1.25 + 4.69i)T + (-1.18e3 + 684.5i)T^{2} \)
41 \( 1 + (-11.2 + 3.01i)T + (1.45e3 - 840.5i)T^{2} \)
43 \( 1 + (3.77 - 2.17i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-33 + 33i)T - 2.20e3iT^{2} \)
53 \( 1 + 3.89T + 2.80e3T^{2} \)
59 \( 1 + (7.16 + 1.91i)T + (3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (-24.6 - 42.7i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-30.1 + 8.08i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + (-13.9 + 52.1i)T + (-4.36e3 - 2.52e3i)T^{2} \)
73 \( 1 + (80.7 - 80.7i)T - 5.32e3iT^{2} \)
79 \( 1 - 110.T + 6.24e3T^{2} \)
83 \( 1 + (-8.16 - 8.16i)T + 6.88e3iT^{2} \)
89 \( 1 + (-22.2 - 82.9i)T + (-6.85e3 + 3.96e3i)T^{2} \)
97 \( 1 + (-8.51 + 31.7i)T + (-8.14e3 - 4.70e3i)T^{2} \)
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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

−16.71487115257033921077039831449, −16.09187475882675765830781327503, −14.90121673873846363130211301342, −13.23088212758030324102416042000, −12.43893292379960252889062441682, −10.76596277264235722033582706737, −9.681118521230812636999806410897, −7.36696718885324987891037216057, −5.46016289798215233036269888470, −3.72142540296874082628417979044, 3.53781147181532421255962996871, 6.07665860600058424037499978925, 7.07571546754710750366822790690, 9.238107072771251347498232119849, 11.18888117674074169664036326344, 12.40595888421063200294191335786, 13.31895380104728831499436591505, 14.82544444620730017196385782624, 15.94149412892433836545463795814, 17.02124909829761653818860132298

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