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Properties

Label 2-11e2-11.5-c1-0-0
Degree $2$
Conductor $121$
Sign $-0.174 - 0.984i$
Analytic cond. $0.966189$
Root an. cond. $0.982949$
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  + (0.618 + 1.90i)2-s + (0.809 + 0.587i)3-s + (−1.61 + 1.17i)4-s + (0.309 − 0.951i)5-s + (−0.618 + 1.90i)6-s + (−1.61 + 1.17i)7-s + (−0.618 − 1.90i)9-s + 1.99·10-s − 1.99·12-s + (−1.23 − 3.80i)13-s + (−3.23 − 2.35i)14-s + (0.809 − 0.587i)15-s + (−1.23 + 3.80i)16-s + (0.618 − 1.90i)17-s + (3.23 − 2.35i)18-s + ⋯
L(s)  = 1  + (0.437 + 1.34i)2-s + (0.467 + 0.339i)3-s + (−0.809 + 0.587i)4-s + (0.138 − 0.425i)5-s + (−0.252 + 0.776i)6-s + (−0.611 + 0.444i)7-s + (−0.206 − 0.634i)9-s + 0.632·10-s − 0.577·12-s + (−0.342 − 1.05i)13-s + (−0.864 − 0.628i)14-s + (0.208 − 0.151i)15-s + (−0.309 + 0.951i)16-s + (0.149 − 0.461i)17-s + (0.762 − 0.554i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.174 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.174 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(121\)    =    \(11^{2}\)
Sign: $-0.174 - 0.984i$
Analytic conductor: \(0.966189\)
Root analytic conductor: \(0.982949\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{121} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 121,\ (\ :1/2),\ -0.174 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.914628 + 1.09063i\)
\(L(\frac12)\) \(\approx\) \(0.914628 + 1.09063i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
good2 \( 1 + (-0.618 - 1.90i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (-0.809 - 0.587i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (-0.309 + 0.951i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (1.61 - 1.17i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (1.23 + 3.80i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-0.618 + 1.90i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (5.87 + 18.0i)T^{2} \)
23 \( 1 + T + 23T^{2} \)
29 \( 1 + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-2.16 - 6.65i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (2.42 - 1.76i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (6.47 + 4.70i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 6T + 43T^{2} \)
47 \( 1 + (6.47 + 4.70i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (1.85 + 5.70i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (4.04 - 2.93i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (3.70 - 11.4i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 7T + 67T^{2} \)
71 \( 1 + (0.927 - 2.85i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-3.23 + 2.35i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-3.09 - 9.51i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-1.85 + 5.70i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 15T + 89T^{2} \)
97 \( 1 + (2.16 + 6.65i)T + (-78.4 + 57.0i)T^{2} \)
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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

−14.03201478379900984676172749211, −13.02181320360794293250751067355, −12.08526684871614625099246303759, −10.35201309558406311667888410672, −9.155632202663895206372875764647, −8.336856758388475125966117588826, −7.04946400316430978006336104836, −5.91649193120471835366207873474, −4.90425149882598123780545724592, −3.22592864038877381696018567953, 2.02866506823839859598279432937, 3.24192834834163559046372978032, 4.59935306323874333126664031857, 6.54419353677406404941339950950, 7.76199415890923757286587926662, 9.321866092154329620502773866796, 10.30764802693525205854483697217, 11.13694118401869825506872316173, 12.19819170035980524795334847302, 13.15775002120475738175601832186

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