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Properties

Label 2-26-13.2-c2-0-2
Degree $2$
Conductor $26$
Sign $-0.137 + 0.990i$
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 + (−2.78 − 4.83i)3-s + (1.73 + i)4-s + (0.323 − 0.323i)5-s + (2.04 + 7.62i)6-s + (7.67 − 2.05i)7-s + (−1.99 − 2i)8-s + (−11.0 + 19.1i)9-s + (−0.560 + 0.323i)10-s + (1.44 − 5.40i)11-s − 11.1i·12-s + (12.8 − 1.93i)13-s − 11.2·14-s + (−2.46 − 0.661i)15-s + (1.99 + 3.46i)16-s + (−1.74 − 1.00i)17-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (−0.929 − 1.61i)3-s + (0.433 + 0.250i)4-s + (0.0647 − 0.0647i)5-s + (0.340 + 1.27i)6-s + (1.09 − 0.293i)7-s + (−0.249 − 0.250i)8-s + (−1.22 + 2.12i)9-s + (−0.0560 + 0.0323i)10-s + (0.131 − 0.491i)11-s − 0.929i·12-s + (0.988 − 0.148i)13-s − 0.803·14-s + (−0.164 − 0.0440i)15-s + (0.124 + 0.216i)16-s + (−0.102 − 0.0591i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.137 + 0.990i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.137 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(26\)    =    \(2 \cdot 13\)
Sign: $-0.137 + 0.990i$
Analytic conductor: \(0.708448\)
Root analytic conductor: \(0.841693\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{26} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 26,\ (\ :1),\ -0.137 + 0.990i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.393059 - 0.451464i\)
\(L(\frac12)\) \(\approx\) \(0.393059 - 0.451464i\)
\(L(2)\) 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 \( 1 + (-12.8 + 1.93i)T \)
good3 \( 1 + (2.78 + 4.83i)T + (-4.5 + 7.79i)T^{2} \)
5 \( 1 + (-0.323 + 0.323i)T - 25iT^{2} \)
7 \( 1 + (-7.67 + 2.05i)T + (42.4 - 24.5i)T^{2} \)
11 \( 1 + (-1.44 + 5.40i)T + (-104. - 60.5i)T^{2} \)
17 \( 1 + (1.74 + 1.00i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-1.19 - 4.47i)T + (-312. + 180.5i)T^{2} \)
23 \( 1 + (15.0 - 8.71i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-8.25 - 14.2i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (9.27 - 9.27i)T - 961iT^{2} \)
37 \( 1 + (-9.12 + 34.0i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 + (-58.6 - 15.7i)T + (1.45e3 + 840.5i)T^{2} \)
43 \( 1 + (45.2 + 26.1i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-8.73 - 8.73i)T + 2.20e3iT^{2} \)
53 \( 1 + 23.4T + 2.80e3T^{2} \)
59 \( 1 + (-34.0 + 9.13i)T + (3.01e3 - 1.74e3i)T^{2} \)
61 \( 1 + (13.9 - 24.1i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (33.0 + 8.85i)T + (3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + (-19.7 - 73.8i)T + (-4.36e3 + 2.52e3i)T^{2} \)
73 \( 1 + (18.9 + 18.9i)T + 5.32e3iT^{2} \)
79 \( 1 + 142.T + 6.24e3T^{2} \)
83 \( 1 + (91.7 - 91.7i)T - 6.88e3iT^{2} \)
89 \( 1 + (-40.4 + 150. i)T + (-6.85e3 - 3.96e3i)T^{2} \)
97 \( 1 + (-29.1 - 108. i)T + (-8.14e3 + 4.70e3i)T^{2} \)
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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

−17.41461812657984184184339060513, −16.27803845652486103237374598531, −14.10306424032303542770968861693, −12.91539288074473566650069626333, −11.57739129092242009499826208260, −10.91021558369131173762645646492, −8.415811805627628070892323830060, −7.32621940736378706558100165065, −5.79575268262008453379640960603, −1.40446580441834285903588825862, 4.51599376519828505595508846068, 6.07070787594747835733965990320, 8.508536336186558123739459273033, 9.849394604783779239736900968255, 10.94924060910041252954711629633, 11.80812928854307069822477711825, 14.49230941755979746750664787309, 15.48711698447618747463853937104, 16.38108074355194681393105864759, 17.51150372100180649902679229878

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