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Properties

Label 2-11e2-11.7-c2-0-4
Degree $2$
Conductor $121$
Sign $-0.822 - 0.568i$
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  + (−1.54 + 4.75i)3-s + (1.23 + 3.80i)4-s + (0.809 − 0.587i)5-s + (−12.9 − 9.40i)9-s − 20.0·12-s + (1.54 + 4.75i)15-s + (−12.9 + 9.40i)16-s + (3.23 + 2.35i)20-s + 35·23-s + (−7.41 + 22.8i)25-s + (28.3 − 20.5i)27-s + (29.9 + 21.7i)31-s + (19.7 − 60.8i)36-s + (−7.72 − 23.7i)37-s − 16.0·45-s + ⋯
L(s)  = 1  + (−0.515 + 1.58i)3-s + (0.309 + 0.951i)4-s + (0.161 − 0.117i)5-s + (−1.43 − 1.04i)9-s − 1.66·12-s + (0.103 + 0.317i)15-s + (−0.809 + 0.587i)16-s + (0.161 + 0.117i)20-s + 1.52·23-s + (−0.296 + 0.913i)25-s + (1.04 − 0.761i)27-s + (0.965 + 0.701i)31-s + (0.549 − 1.69i)36-s + (−0.208 − 0.642i)37-s − 0.355·45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.822 - 0.568i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.822 - 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(121\)    =    \(11^{2}\)
Sign: $-0.822 - 0.568i$
Analytic conductor: \(3.29701\)
Root analytic conductor: \(1.81576\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{121} (40, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 121,\ (\ :1),\ -0.822 - 0.568i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.344695 + 1.10540i\)
\(L(\frac12)\) \(\approx\) \(0.344695 + 1.10540i\)
\(L(2)\) 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 \( 1 + (-1.23 - 3.80i)T^{2} \)
3 \( 1 + (1.54 - 4.75i)T + (-7.28 - 5.29i)T^{2} \)
5 \( 1 + (-0.809 + 0.587i)T + (7.72 - 23.7i)T^{2} \)
7 \( 1 + (39.6 - 28.8i)T^{2} \)
13 \( 1 + (-52.2 - 160. i)T^{2} \)
17 \( 1 + (-89.3 + 274. i)T^{2} \)
19 \( 1 + (292. + 212. i)T^{2} \)
23 \( 1 - 35T + 529T^{2} \)
29 \( 1 + (680. - 494. i)T^{2} \)
31 \( 1 + (-29.9 - 21.7i)T + (296. + 913. i)T^{2} \)
37 \( 1 + (7.72 + 23.7i)T + (-1.10e3 + 804. i)T^{2} \)
41 \( 1 + (1.35e3 + 988. i)T^{2} \)
43 \( 1 - 1.84e3T^{2} \)
47 \( 1 + (-15.4 + 47.5i)T + (-1.78e3 - 1.29e3i)T^{2} \)
53 \( 1 + (-56.6 - 41.1i)T + (868. + 2.67e3i)T^{2} \)
59 \( 1 + (-33.0 - 101. i)T + (-2.81e3 + 2.04e3i)T^{2} \)
61 \( 1 + (-1.14e3 + 3.53e3i)T^{2} \)
67 \( 1 - 35T + 4.48e3T^{2} \)
71 \( 1 + (-107. + 78.1i)T + (1.55e3 - 4.79e3i)T^{2} \)
73 \( 1 + (4.31e3 - 3.13e3i)T^{2} \)
79 \( 1 + (-1.92e3 - 5.93e3i)T^{2} \)
83 \( 1 + (-2.12e3 + 6.55e3i)T^{2} \)
89 \( 1 + 97T + 7.92e3T^{2} \)
97 \( 1 + (76.8 + 55.8i)T + (2.90e3 + 8.94e3i)T^{2} \)
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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

−13.59541425989725806029066661578, −12.38991991565763977785784604262, −11.37882812272856355692687300821, −10.64286242985979809184059678781, −9.459600961027247287272221511528, −8.575758025858127125460697477586, −7.02131760273002703657862122278, −5.47925354592088601042589981845, −4.31023581242658976852175392706, −3.12742100995012983515030620297, 0.900816192103867548833522817947, 2.34780680738385990900557693048, 5.16542847904964446427952040683, 6.30987815551678861426062510053, 6.94953653815952366472027852881, 8.232442790498313527199037882306, 9.777572763632633119776009695599, 11.00966173014989326776836573355, 11.75227161628818607158666930828, 12.85095810156682390205527323892

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