Properties

Label 2-11-11.10-c14-0-8
Degree $2$
Conductor $11$
Sign $1$
Analytic cond. $13.6761$
Root an. cond. $3.69813$
Motivic weight $14$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.51e3·3-s + 1.63e4·4-s + 1.00e5·5-s + 1.54e6·9-s − 1.94e7·11-s + 4.12e7·12-s + 2.53e8·15-s + 2.68e8·16-s + 1.65e9·20-s + 1.55e9·23-s + 4.05e9·25-s − 8.15e9·27-s − 5.34e10·31-s − 4.90e10·33-s + 2.52e10·36-s + 1.26e11·37-s − 3.19e11·44-s + 1.55e11·45-s + 7.15e11·47-s + 6.75e11·48-s + 6.78e11·49-s − 2.21e12·53-s − 1.96e12·55-s − 4.95e12·59-s + 4.15e12·60-s + 4.39e12·64-s − 1.16e13·67-s + ⋯
L(s)  = 1  + 1.14·3-s + 4-s + 1.29·5-s + 0.322·9-s − 11-s + 1.14·12-s + 1.48·15-s + 16-s + 1.29·20-s + 0.457·23-s + 0.664·25-s − 0.779·27-s − 1.94·31-s − 1.14·33-s + 0.322·36-s + 1.33·37-s − 44-s + 0.416·45-s + 1.41·47-s + 1.14·48-s + 49-s − 1.88·53-s − 1.29·55-s − 1.99·59-s + 1.48·60-s + 64-s − 1.92·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(11\)
Sign: $1$
Analytic conductor: \(13.6761\)
Root analytic conductor: \(3.69813\)
Motivic weight: \(14\)
Rational: yes
Arithmetic: yes
Character: $\chi_{11} (10, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 11,\ (\ :7),\ 1)\)

Particular Values

\(L(\frac{15}{2})\) \(\approx\) \(3.651430639\)
\(L(\frac12)\) \(\approx\) \(3.651430639\)
\(L(8)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + p^{7} T \)
good2 \( ( 1 - p^{7} T )( 1 + p^{7} T ) \)
3 \( 1 - 2515 T + p^{14} T^{2} \)
5 \( 1 - 100799 T + p^{14} T^{2} \)
7 \( ( 1 - p^{7} T )( 1 + p^{7} T ) \)
13 \( ( 1 - p^{7} T )( 1 + p^{7} T ) \)
17 \( ( 1 - p^{7} T )( 1 + p^{7} T ) \)
19 \( ( 1 - p^{7} T )( 1 + p^{7} T ) \)
23 \( 1 - 1556561195 T + p^{14} T^{2} \)
29 \( ( 1 - p^{7} T )( 1 + p^{7} T ) \)
31 \( 1 + 53499153997 T + p^{14} T^{2} \)
37 \( 1 - 126509871575 T + p^{14} T^{2} \)
41 \( ( 1 - p^{7} T )( 1 + p^{7} T ) \)
43 \( ( 1 - p^{7} T )( 1 + p^{7} T ) \)
47 \( 1 - 715778884850 T + p^{14} T^{2} \)
53 \( 1 + 2217378708790 T + p^{14} T^{2} \)
59 \( 1 + 4954816467613 T + p^{14} T^{2} \)
61 \( ( 1 - p^{7} T )( 1 + p^{7} T ) \)
67 \( 1 + 11652648832405 T + p^{14} T^{2} \)
71 \( 1 - 14633687116307 T + p^{14} T^{2} \)
73 \( ( 1 - p^{7} T )( 1 + p^{7} T ) \)
79 \( ( 1 - p^{7} T )( 1 + p^{7} T ) \)
83 \( ( 1 - p^{7} T )( 1 + p^{7} T ) \)
89 \( 1 + 68889823168417 T + p^{14} T^{2} \)
97 \( 1 - 68911099629215 T + p^{14} T^{2} \)
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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

−16.88548555504378258966854113605, −15.34889424288392848920080124965, −14.15364067490745431664172450693, −12.91916212939582220070254328702, −10.75988742334461778376489929962, −9.319132507906152340943483480990, −7.61617666453320400902485672995, −5.79536832218465334955958767270, −2.90121695774692269707827184293, −1.89040868250106846753560087550, 1.89040868250106846753560087550, 2.90121695774692269707827184293, 5.79536832218465334955958767270, 7.61617666453320400902485672995, 9.319132507906152340943483480990, 10.75988742334461778376489929962, 12.91916212939582220070254328702, 14.15364067490745431664172450693, 15.34889424288392848920080124965, 16.88548555504378258966854113605

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