Properties

Label 2-26-13.7-c2-0-0
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: $\chi_{26} (7, \cdot )$
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

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

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