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Properties

Label 2-11e2-11.4-c1-0-3
Degree $2$
Conductor $121$
Sign $0.859 + 0.511i$
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  + (0.809 + 0.587i)2-s + (0.618 − 1.90i)3-s + (−0.309 − 0.951i)4-s + (−0.809 + 0.587i)5-s + (1.61 − 1.17i)6-s + (0.618 + 1.90i)7-s + (0.927 − 2.85i)8-s + (−0.809 − 0.587i)9-s − 10-s − 1.99·12-s + (0.809 + 0.587i)13-s + (−0.618 + 1.90i)14-s + (0.618 + 1.90i)15-s + (0.809 − 0.587i)16-s + (−4.04 + 2.93i)17-s + (−0.309 − 0.951i)18-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (0.356 − 1.09i)3-s + (−0.154 − 0.475i)4-s + (−0.361 + 0.262i)5-s + (0.660 − 0.479i)6-s + (0.233 + 0.718i)7-s + (0.327 − 1.00i)8-s + (−0.269 − 0.195i)9-s − 0.316·10-s − 0.577·12-s + (0.224 + 0.163i)13-s + (−0.165 + 0.508i)14-s + (0.159 + 0.491i)15-s + (0.202 − 0.146i)16-s + (−0.981 + 0.712i)17-s + (−0.0728 − 0.224i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(121\)    =    \(11^{2}\)
Sign: $0.859 + 0.511i$
Analytic conductor: \(0.966189\)
Root analytic conductor: \(0.982949\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{121} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 121,\ (\ :1/2),\ 0.859 + 0.511i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36911 - 0.376555i\)
\(L(\frac12)\) \(\approx\) \(1.36911 - 0.376555i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.809 - 0.587i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (-0.618 + 1.90i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (0.809 - 0.587i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (-0.618 - 1.90i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-0.809 - 0.587i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (4.04 - 2.93i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.85 - 5.70i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 2T + 23T^{2} \)
29 \( 1 + (2.78 + 8.55i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-1.61 - 1.17i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.927 + 2.85i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.54 + 4.75i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + (-0.618 + 1.90i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (7.28 + 5.29i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-2.47 - 7.60i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-4.85 + 3.52i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 2T + 67T^{2} \)
71 \( 1 + (9.70 - 7.05i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-0.618 - 1.90i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (8.09 + 5.87i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-4.85 + 3.52i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 9T + 89T^{2} \)
97 \( 1 + (-10.5 - 7.64i)T + (29.9 + 92.2i)T^{2} \)
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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

−13.36698931711887760339571072178, −12.72528376820201862976593573944, −11.58306610875824675153532381267, −10.29843181471543889399007878186, −8.854029960385424933647284807661, −7.75384528321022666444419855021, −6.64579030538668847188487540763, −5.70061139574055580956689523721, −4.06639688122683709048418171842, −1.92659410493947734409567211885, 3.01375699281306565671805504984, 4.27241533587434742940651276809, 4.82177005491798371864729318047, 7.07909311579771284569498995016, 8.436618857388649544125903002040, 9.287048105505832980344628472181, 10.70489484640390481652921542669, 11.34616612103008001366064837771, 12.65465774129199013639471913502, 13.51354300739221769935451639042

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