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Properties

Label 2-26-13.9-c1-0-0
Degree $2$
Conductor $26$
Sign $0.859 - 0.511i$
Analytic cond. $0.207611$
Root an. cond. $0.455643$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s − 5-s + (−2 − 3.46i)7-s + 0.999·8-s + (1.5 + 2.59i)9-s + (0.5 − 0.866i)10-s + (−2 + 3.46i)11-s + (3.5 + 0.866i)13-s + 3.99·14-s + (−0.5 + 0.866i)16-s + (−1.5 − 2.59i)17-s − 3·18-s + (0.499 + 0.866i)20-s + (−1.99 − 3.46i)22-s + (2 − 3.46i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s − 0.447·5-s + (−0.755 − 1.30i)7-s + 0.353·8-s + (0.5 + 0.866i)9-s + (0.158 − 0.273i)10-s + (−0.603 + 1.04i)11-s + (0.970 + 0.240i)13-s + 1.06·14-s + (−0.125 + 0.216i)16-s + (−0.363 − 0.630i)17-s − 0.707·18-s + (0.111 + 0.193i)20-s + (−0.426 − 0.738i)22-s + (0.417 − 0.722i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(26\)    =    \(2 \cdot 13\)
Sign: $0.859 - 0.511i$
Analytic conductor: \(0.207611\)
Root analytic conductor: \(0.455643\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{26} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 26,\ (\ :1/2),\ 0.859 - 0.511i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.518211 + 0.142421i\)
\(L(\frac12)\) \(\approx\) \(0.518211 + 0.142421i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (-3.5 - 0.866i)T \)
good3 \( 1 + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + T + 5T^{2} \)
7 \( 1 + (2 + 3.46i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2 - 3.46i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + (1.5 - 2.59i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.5 + 7.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4 - 6.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 8T + 47T^{2} \)
53 \( 1 + 9T + 53T^{2} \)
59 \( 1 + (-2 - 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.5 + 6.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2 - 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4 - 6.92i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 11T + 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1 + 1.73i)T + (-48.5 + 84.0i)T^{2} \)
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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

−17.50799021433546123665920387934, −16.26623720731594403394103939384, −15.64886343391872673852350459186, −13.90625455460884407486336769723, −12.96336081772086273433631127787, −10.86528555775159191282929273534, −9.795272204285249139218264653633, −7.87360442404991356808984330857, −6.81910561654721702742782169449, −4.44784784558725094005357709970, 3.37568655586676820268862357978, 6.08899132613771253922925065251, 8.321664699454448960348307050955, 9.455354274309319704863065403827, 11.07879755217083448900979344953, 12.28901214179054392561517743721, 13.30524454945633262033065358911, 15.36446925692976134734598803910, 16.02911031406074142986274222501, 17.83917270532275816683966581168

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