Properties

Label 2-26-13.12-c5-0-0
Degree $2$
Conductor $26$
Sign $-0.192 - 0.981i$
Analytic cond. $4.16997$
Root an. cond. $2.04205$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4i·2-s − 13·3-s − 16·4-s + 51i·5-s + 52i·6-s + 105i·7-s + 64i·8-s − 74·9-s + 204·10-s + 120i·11-s + 208·12-s + (−598 + 117i)13-s + 420·14-s − 663i·15-s + 256·16-s − 1.10e3·17-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.833·3-s − 0.5·4-s + 0.912i·5-s + 0.589i·6-s + 0.809i·7-s + 0.353i·8-s − 0.304·9-s + 0.645·10-s + 0.299i·11-s + 0.416·12-s + (−0.981 + 0.192i)13-s + 0.572·14-s − 0.760i·15-s + 0.250·16-s − 0.923·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.192 - 0.981i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.192 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(26\)    =    \(2 \cdot 13\)
Sign: $-0.192 - 0.981i$
Analytic conductor: \(4.16997\)
Root analytic conductor: \(2.04205\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{26} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 26,\ (\ :5/2),\ -0.192 - 0.981i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.326068 + 0.396047i\)
\(L(\frac12)\) \(\approx\) \(0.326068 + 0.396047i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 4iT \)
13 \( 1 + (598 - 117i)T \)
good3 \( 1 + 13T + 243T^{2} \)
5 \( 1 - 51iT - 3.12e3T^{2} \)
7 \( 1 - 105iT - 1.68e4T^{2} \)
11 \( 1 - 120iT - 1.61e5T^{2} \)
17 \( 1 + 1.10e3T + 1.41e6T^{2} \)
19 \( 1 + 1.17e3iT - 2.47e6T^{2} \)
23 \( 1 - 1.05e3T + 6.43e6T^{2} \)
29 \( 1 + 4.10e3T + 2.05e7T^{2} \)
31 \( 1 - 9.62e3iT - 2.86e7T^{2} \)
37 \( 1 - 8.70e3iT - 6.93e7T^{2} \)
41 \( 1 + 9.48e3iT - 1.15e8T^{2} \)
43 \( 1 - 9.99e3T + 1.47e8T^{2} \)
47 \( 1 + 2.94e3iT - 2.29e8T^{2} \)
53 \( 1 + 750T + 4.18e8T^{2} \)
59 \( 1 + 4.09e4iT - 7.14e8T^{2} \)
61 \( 1 + 5.79e4T + 8.44e8T^{2} \)
67 \( 1 - 2.28e4iT - 1.35e9T^{2} \)
71 \( 1 - 6.37e4iT - 1.80e9T^{2} \)
73 \( 1 - 5.88e4iT - 2.07e9T^{2} \)
79 \( 1 - 6.32e4T + 3.07e9T^{2} \)
83 \( 1 - 5.54e4iT - 3.93e9T^{2} \)
89 \( 1 + 1.04e5iT - 5.58e9T^{2} \)
97 \( 1 - 1.60e5iT - 8.58e9T^{2} \)
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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

−17.15032487127145858454802249843, −15.41651347031030249678046257688, −14.26106398043829338349800275167, −12.56369811219189008281294237284, −11.52082116997537437532205270568, −10.59616699188465335893428155325, −9.019352864244756252791107995483, −6.80157818406843141459528053403, −5.09729398035414979897109511950, −2.64216880638013891614027301533, 0.36804425174519619519079332654, 4.56480353252279644550751696054, 5.92299613080343456668111599202, 7.61164497672029724551853436436, 9.199627924773101230209134670300, 10.86587694324505115992614974363, 12.35847631125845181255653492078, 13.54703573821438560300410697688, 14.97850302551614018557915750027, 16.70352380946226187145853058933

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