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Properties

Label 2-11e2-11.7-c2-0-7
Degree $2$
Conductor $121$
Sign $0.999 - 0.0237i$
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  + (0.427 − 0.587i)2-s + (0.427 − 1.31i)3-s + (1.07 + 3.30i)4-s + (−3.23 + 2.35i)5-s + (−0.590 − 0.812i)6-s + (9.47 − 3.07i)7-s + (5.16 + 1.67i)8-s + (5.73 + 4.16i)9-s + 2.90i·10-s + 4.79·12-s + (3.09 − 4.25i)13-s + (2.23 − 6.88i)14-s + (1.70 + 5.25i)15-s + (−8.04 + 5.84i)16-s + (−8.98 − 12.3i)17-s + (4.89 − 1.59i)18-s + ⋯
L(s)  = 1  + (0.213 − 0.293i)2-s + (0.142 − 0.438i)3-s + (0.268 + 0.825i)4-s + (−0.647 + 0.470i)5-s + (−0.0983 − 0.135i)6-s + (1.35 − 0.439i)7-s + (0.645 + 0.209i)8-s + (0.637 + 0.463i)9-s + 0.290i·10-s + 0.399·12-s + (0.237 − 0.327i)13-s + (0.159 − 0.491i)14-s + (0.113 + 0.350i)15-s + (−0.502 + 0.365i)16-s + (−0.528 − 0.727i)17-s + (0.272 − 0.0884i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0237i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0237i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(121\)    =    \(11^{2}\)
Sign: $0.999 - 0.0237i$
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: \((2,\ 121,\ (\ :1),\ 0.999 - 0.0237i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.78490 + 0.0212180i\)
\(L(\frac12)\) \(\approx\) \(1.78490 + 0.0212180i\)
\(L(2)\) 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 \( 1 + (-0.427 + 0.587i)T + (-1.23 - 3.80i)T^{2} \)
3 \( 1 + (-0.427 + 1.31i)T + (-7.28 - 5.29i)T^{2} \)
5 \( 1 + (3.23 - 2.35i)T + (7.72 - 23.7i)T^{2} \)
7 \( 1 + (-9.47 + 3.07i)T + (39.6 - 28.8i)T^{2} \)
13 \( 1 + (-3.09 + 4.25i)T + (-52.2 - 160. i)T^{2} \)
17 \( 1 + (8.98 + 12.3i)T + (-89.3 + 274. i)T^{2} \)
19 \( 1 + (-1.97 - 0.640i)T + (292. + 212. i)T^{2} \)
23 \( 1 + 2.76T + 529T^{2} \)
29 \( 1 + (27.0 - 8.78i)T + (680. - 494. i)T^{2} \)
31 \( 1 + (-5.76 - 4.18i)T + (296. + 913. i)T^{2} \)
37 \( 1 + (12.4 + 38.2i)T + (-1.10e3 + 804. i)T^{2} \)
41 \( 1 + (66.7 + 21.6i)T + (1.35e3 + 988. i)T^{2} \)
43 \( 1 + 23.0iT - 1.84e3T^{2} \)
47 \( 1 + (-8.41 + 25.9i)T + (-1.78e3 - 1.29e3i)T^{2} \)
53 \( 1 + (-9.27 - 6.73i)T + (868. + 2.67e3i)T^{2} \)
59 \( 1 + (0.701 + 2.15i)T + (-2.81e3 + 2.04e3i)T^{2} \)
61 \( 1 + (13.2 + 18.2i)T + (-1.14e3 + 3.53e3i)T^{2} \)
67 \( 1 + 38.4T + 4.48e3T^{2} \)
71 \( 1 + (61.7 - 44.8i)T + (1.55e3 - 4.79e3i)T^{2} \)
73 \( 1 + (-97.8 + 31.7i)T + (4.31e3 - 3.13e3i)T^{2} \)
79 \( 1 + (-2.31 + 3.18i)T + (-1.92e3 - 5.93e3i)T^{2} \)
83 \( 1 + (48.9 + 67.3i)T + (-2.12e3 + 6.55e3i)T^{2} \)
89 \( 1 - 123.T + 7.92e3T^{2} \)
97 \( 1 + (-62.6 - 45.4i)T + (2.90e3 + 8.94e3i)T^{2} \)
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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

−13.24608430721444389626932207292, −12.08507079366867590106096304464, −11.27724857609205320768454040198, −10.56473282727290380683008345857, −8.585641832430193094422436579077, −7.57332058351616874710989306041, −7.15936347847098095967860754134, −4.87542967611227668098779606388, −3.66565193744783094166450421694, −1.95421723528548266955193526224, 1.58926618905434913002128639065, 4.18865634884273664153189699015, 5.02231804911689194350855251555, 6.44483216941428225565452982395, 7.85364584199673961171463119531, 8.897491290598051814394694488016, 10.12606350737854798136304936324, 11.20012016548317624000201222611, 12.00639009456267681314889007728, 13.38430357870330095292307948636

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