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Properties

Label 2-11e2-11.10-c2-0-0
Degree $2$
Conductor $121$
Sign $0.522 - 0.852i$
Analytic cond. $3.29701$
Root an. cond. $1.81576$
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  − 2.89i·2-s − 3.85·3-s − 4.35·4-s − 1.96·5-s + 11.1i·6-s + 9.76i·7-s + 1.02i·8-s + 5.84·9-s + 5.68i·10-s + 16.7·12-s + 3.56i·13-s + 28.2·14-s + 7.58·15-s − 14.4·16-s + 4.57i·17-s − 16.8i·18-s + ⋯
L(s)  = 1  − 1.44i·2-s − 1.28·3-s − 1.08·4-s − 0.393·5-s + 1.85i·6-s + 1.39i·7-s + 0.128i·8-s + 0.648·9-s + 0.568i·10-s + 1.39·12-s + 0.274i·13-s + 2.01·14-s + 0.505·15-s − 0.903·16-s + 0.269i·17-s − 0.937i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(121\)    =    \(11^{2}\)
Sign: $0.522 - 0.852i$
Analytic conductor: \(3.29701\)
Root analytic conductor: \(1.81576\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{121} (120, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 121,\ (\ :1),\ 0.522 - 0.852i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.221816 + 0.124268i\)
\(L(\frac12)\) \(\approx\) \(0.221816 + 0.124268i\)
\(L(2)\) 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 \( 1 + 2.89iT - 4T^{2} \)
3 \( 1 + 3.85T + 9T^{2} \)
5 \( 1 + 1.96T + 25T^{2} \)
7 \( 1 - 9.76iT - 49T^{2} \)
13 \( 1 - 3.56iT - 169T^{2} \)
17 \( 1 - 4.57iT - 289T^{2} \)
19 \( 1 - 21.8iT - 361T^{2} \)
23 \( 1 + 35.9T + 529T^{2} \)
29 \( 1 - 6.35iT - 841T^{2} \)
31 \( 1 - 4.74T + 961T^{2} \)
37 \( 1 + 48.8T + 1.36e3T^{2} \)
41 \( 1 + 55.1iT - 1.68e3T^{2} \)
43 \( 1 - 35.4iT - 1.84e3T^{2} \)
47 \( 1 + 50.1T + 2.20e3T^{2} \)
53 \( 1 - 21.9T + 2.80e3T^{2} \)
59 \( 1 - 32.5T + 3.48e3T^{2} \)
61 \( 1 + 79.0iT - 3.72e3T^{2} \)
67 \( 1 - 33.2T + 4.48e3T^{2} \)
71 \( 1 - 94.4T + 5.04e3T^{2} \)
73 \( 1 + 14.3iT - 5.32e3T^{2} \)
79 \( 1 + 19.7iT - 6.24e3T^{2} \)
83 \( 1 - 63.1iT - 6.88e3T^{2} \)
89 \( 1 - 42.3T + 7.92e3T^{2} \)
97 \( 1 + 176.T + 9.40e3T^{2} \)
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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

−12.59377129651607352902443414547, −12.06177291127578015683348931330, −11.60712940294158552650545652128, −10.58325523647040671334558607447, −9.636876373270273925262241873490, −8.308108670293838568406753742557, −6.37340842317533677756869999361, −5.31933521858444548671833382998, −3.78389576184685500767964558087, −1.98323657284832484141083249494, 0.20160638829453102279879052542, 4.23666916720942213439806400703, 5.33453992190262604639676245035, 6.48213273525139790866952780320, 7.25149093886219554957534412632, 8.260789798548531935623923759706, 9.972748272155117400301994997587, 11.05698095054419000827943448015, 11.88379738317534104239842458579, 13.37581692521190538925069195326

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